1686:
454:
445:
31:
7619:
1659:
6560:
6342:
6353:
7688:, and vertices are the leaf nodes. A hypergraph is then just a collection of trees with common, shared nodes (that is, a given internal node or leaf may occur in several different trees). Conversely, every collection of trees can be understood as this generalized hypergraph. Since trees are widely used throughout
4601:
1717:
An alternative representation of the hypergraph called PAOH is shown in the figure on top of this article. Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges. It has been designed for dynamic hypergraphs but can be used
1797:
There are many generalizations of classic hypergraph coloring. One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Some mixed hypergraphs are uncolorable for any number of colors. A general criterion for uncolorability is unknown. When a mixed hypergraph is
1693:
In another style of hypergraph visualization, the subdivision model of hypergraph drawing, the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. The hyperedges of the hypergraph are represented by contiguous subsets of these regions, which may be indicated
5188:
defined the stronger notions of β-acyclicity and γ-acyclicity. We can state β-acyclicity as the requirement that all subhypergraphs of the hypergraph are α-acyclic, which is equivalent to an earlier definition by Graham. The notion of γ-acyclicity is a more restrictive condition which is equivalent
1677:
style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves. If the vertices are represented as points, the hyperedges may also be shown as smooth
1782:
to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. In other words, there must be no monochromatic hyperedge with cardinality at least 2. In this sense it is a direct generalization of graph coloring. Minimum number of used distinct
7971:
Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. This allows graphs with edge-loops, which need not contain vertices at all. For example, consider the generalized hypergraph consisting of two edges
5183:
Note that α-acyclicity has the counter-intuitive property that adding hyperedges to an α-cyclic hypergraph may make it α-acyclic (for instance, adding a hyperedge containing all vertices of the hypergraph will always make it α-acyclic). Motivated in part by this perceived shortcoming,
4960:
7679:
One possible generalization of a hypergraph is to allow edges to point at other edges. There are two variations of this generalization. In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, subsets of subsets of vertices and so on
3883:
6135:
6555:{\displaystyle G=\lbrace f_{1}=\lbrace \alpha ,\beta \rbrace ,f_{2}=\lbrace \beta ,\gamma \rbrace ,f_{3}=\lbrace \gamma ,\delta \rbrace ,f_{4}=\lbrace \delta ,\alpha \rbrace ,f_{5}=\lbrace \alpha ,\gamma \rbrace ,f_{6}=\lbrace \beta ,\delta \rbrace \rbrace }
8122:. In particular, there is no transitive closure of set membership for such hypergraphs. Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer
5140:
We can define a weaker notion of hypergraph acyclicity, later termed α-acyclicity. This notion of acyclicity is equivalent to the hypergraph being conformal (every clique of the primal graph is covered by some hyperedge) and its primal graph being
4414:
448:
Alternative representation of the hypergraph reported in the figure above, called PAOH. Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertices are aligned to the left. The legend on the right shows the names of the
3173:
2599:
2381:
5200:
Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. However, none of the reverse implications hold, so those four notions are different.
1713:
to determine whether a hypergraph has a planar subdivision drawing, but the existence of a drawing of this type may be tested efficiently when the adjacency pattern of the regions is constrained to be a path, cycle, or tree.
2907:
4811:
3035:
3370:
149:
7550:
6750:
4224:
1359:) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element. Hence any standard graph theoretic concept that is independent of the edge orders
9811:
We next state a theorem due to Elayne Dauber whose corollaries describe properties of line-symmetric graphs. Note the obvious but important observation that every line-symmetric graph is line-regular.
1705:
hyperedges (the curves defining the diagram) and 2 − 1 vertices (represented by the regions into which these curves subdivide the plane). In contrast with the polynomial-time recognition of
3760:
2273:
6835:
2835:
611:
3604:
2782:
3678:
6337:{\displaystyle H=\lbrace e_{1}=\lbrace a,b\rbrace ,e_{2}=\lbrace b,c\rbrace ,e_{3}=\lbrace c,d\rbrace ,e_{4}=\lbrace d,a\rbrace ,e_{5}=\lbrace b,d\rbrace ,e_{6}=\lbrace a,c\rbrace \rbrace }
230:
4669:
Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. This bipartite graph is also called
7350:
4804:
3431:
8133:
The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges. Thus, for the above example, the
6068:
5856:
5465:
294:
1780:
403:
3982:
3292:
3259:
1456:
can be regarded as the incidence graph of a hypergraph when it is 2-colored and it is indicated which color class corresponds to hypergraph vertices and which to hypergraph edges.
5131:
7211:
5792:
4140:
810:
7848:
5561:
862:
517:
6944:
6638:
5928:
660:
342:
8116:
8070:
7921:
5089:
2948:
752:
706:
7796:
7466:
5052:
5021:
1298:
5358:
4977:, there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs.
4596:{\displaystyle b_{ij}=\left\{{\begin{matrix}-1&\mathrm {if} ~v_{i}\in T(e_{j})\\1&\mathrm {if} ~v_{i}\in H(e_{j})\\0&\mathrm {otherwise} .\end{matrix}}\right.}
3064:
2502:
1357:
1331:
4762:
438:
7954:
7881:
7104:
4729:
4406:
3752:
1039:
7751:
7407:
6904:
5888:
5720:
5521:
5320:
3704:
2706:
5639:
4364:
4328:
4262:
1689:
An order-4 Venn diagram, which can be interpreted as a subdivision drawing of a hypergraph with 15 vertices (the 15 colored regions) and 4 hyperedges (the 4 ellipses).
7279:
5294:
5252:
2209:
2076:
1230:
8949:
8194:
7692:
and many other branches of mathematics, one could say that hypergraphs appear naturally as well. So, for example, this generalization arises naturally as a model of
3076:
2510:
8024:
7997:
7434:
7158:
7131:
5740:
5488:
4292:
4095:
4064:
4033:
3916:
3206:
2667:
2476:
2438:
2411:
1425:
1145:
925:
7582:
6988:
5951:
5384:
2287:
6017:
1387:
1666:
can be interpreted as a drawing of a hypergraph in which four vertices (depicted as white rectangles and disks) are connected by three hyperedges drawn as trees.
7373:
7008:
6878:
6858:
6773:
6682:
6662:
6603:
6583:
6127:
5991:
5971:
5694:
5670:
5613:
5589:
5404:
4006:
3226:
2968:
2640:
2097:
2048:
2024:
2002:
1979:
1957:
1897:
1877:
1855:
1189:
1165:
1097:
1069:
1001:
981:
945:
8969:
Tian, Ze; Hwang, TaeHyun; Kuang, Rui (2009), "A hypergraph-based learning algorithm for classifying gene expression and arrayCGH data with prior knowledge",
10528:
9624:
9670:
1670:
Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.
1550:
that introduces extra hypergraph structural cost to restrict the learning results. For large scale hypergraphs, a distributed framework built using
9916:
9860:
8365:
9709:(1984). "Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs".
5165:
enjoys certain desirable properties if its underlying hypergraph is α-acyclic. Besides, α-acyclicity is also related to the expressiveness of the
3521:) of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.
10160:
9232:
1828:- has loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices.
10336:
4955:{\displaystyle a_{ij}=\left\{{\begin{matrix}w_{e_{k}}&\mathrm {if} ~(v_{i},v_{j})\in E\\0&\mathrm {otherwise} .\end{matrix}}\right.}
9213:
8668:
2840:
9823:
Karypis, G., Aggarwal, R., Kumar, V., and
Shekhar, S. (March 1999), "Multilevel hypergraph partitioning: applications in VLSI domain",
9945:
Catalyurek, U.V.; Aykanat, C. (1999), "Hypergraph-Partitioning Based
Decomposition for Parallel Sparse-Matrix Vector Multiplication",
9895:
9062:
2976:
9183:
3300:
1913:
in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. An alternative term is
1427:
is an undirected graph whose edges connect not just two vertices, but an arbitrary number. An undirected hypergraph is also called a
37:
8617:
9431:
9362:
9288:
9163:
2145:: for any two hyperedges, either they are disjoint, or one is included in the other. In other words, the set of hyperedges forms a
440:. This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors.
9430:
Buchin, Kevin; van
Kreveld, Marc; Meijer, Henk; Speckmann, Bettina; Verbeek, Kevin (2010), "On planar supports for hypergraphs",
8251:
7474:
6690:
8552:
8341:
4145:
3878:{\displaystyle b_{ij}=\left\{{\begin{matrix}1&\mathrm {if} ~v_{i}\in e_{j}\\0&\mathrm {otherwise} .\end{matrix}}\right.}
10099:
10078:
10057:
10018:
9997:
9793:
9451:
9382:
9341:
9308:
9177:
8662:
8603:
8446:
2217:
1798:
colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively.
9799:
8452:
6781:
6077:
is an isomorphism from a vertex set into itself, that is a relabeling of vertices. The set of automorphisms of a hypergraph
9044:
Ranshous, Stephen; Joslyn, Cliff; Kreyling, Sean; Nowak, Kathleen; Samatova, Nagiza; West, Curtis; Winters, Samuel (2017).
8905:
8728:
Ghoshal, Gourab; Zlatic, Vinko; Caldarelli, Guido; Newman, Mark E.J. (2009), "Random
Hypergraphs and their applications",
8708:
8488:
4685:
of a graph. In the case of a graph, the adjacency matrix is a square matrix which indicates whether pairs of vertices are
522:
10153:
8398:
2118:- every subset of an undirected hypergraph's edges is a hyperedge too. A downward-closed hypergraph is usually called an
10497:
1514:
Undirected hypergraphs are useful in modelling such things as satisfiability problems, databases, machine learning, and
10543:
10487:
10329:
9475:
3532:
3475:
1639:
10457:
3609:
2787:
2112:
of a hypergraph is the reduced hypergraph obtained by removing every hyperedge which is included in another hyperedge.
1566:
nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.
9686:
9588:
9556:
9169:
8702:
7666:
3375:
When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an
7648:
2108:: no hyperedge is a strict subset of another hyperedge; equivalently, every hyperedge is maximal for inclusion. The
154:
10595:
10254:
8860:
Patro, Rob; Kingsoford, Carl (2013), "Predicting protein interactions via parsimonious network history inference",
10533:
2158:
Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called
1627:
10564:
7290:
5150:
7235:
Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.
10146:
7644:
4767:
3385:
7228:
if all edges are symmetric. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply
6025:
5803:
5412:
1554:
is also available. It can be desirable to study hypergraphs where all hyperedges have the same cardinality; a
234:
10507:
10322:
10038:
9605:
2714:
17:
4992:
defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair
1729:
346:
10538:
10523:
9328:, Lecture Notes in Computer Science, vol. 10439, Springer International Publishing, pp. 387–394,
4686:
3925:
3264:
3231:
1538:(biochemical interactions as hyperedges). Representative hypergraph learning techniques include hypergraph
1523:
877:
8834:
Liu, Qingshan; Huang, Yuchi; Metaxas, Dimitris N. (2013), "Hypergraph with sampling for image retrieval",
7010:
is the maximum cardinality of any of the edges in the hypergraph. If all edges have the same cardinality
10033:
7640:
2120:
1599:
7167:
5748:
4100:
756:
10181:
7801:
5526:
1633:
1609:
814:
460:
6909:
6608:
5893:
5213:
is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.
615:
298:
10590:
10070:
Coloring Mixed
Hypergraphs: Theory, Algorithms and Applications: Theory, Algorithms, and Applications
9932:
A Hypergraph Model for
Mapping Repeated Sparse Matrix–Vector Product Computations onto Multicomputers
9205:
8075:
8029:
1562:. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting
9837:
8285:
Valdivia, Paola; Buono, Paolo; Plaisant, Catherine; Dufournaud, Nicole; Fekete, Jean-Daniel (2020).
7886:
7711:, since it is not transitive. The graph corresponding to the Levi graph of this generalization is a
2920:
710:
664:
10439:
10259:
9959:
8479:. 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). IEEE. pp. 497–508.
7756:
7629:
7439:
3376:
1547:
1235:
9324:
Naheed Anjum, Arafat; Bressan, Stéphane (2017), "Hypergraph
Drawing by Force-Directed Placement",
5331:
3043:
2481:
1336:
1310:
10472:
9403:
7633:
4734:
1593:
1472:
407:
10028:
8642:
7926:
7853:
7074:
5094:
4692:
4369:
3715:
1006:
9954:
9832:
7718:
7712:
7703:
For such a hypergraph, set membership then provides an ordering, but the ordering is neither a
7685:
7378:
6883:
5867:
5699:
5500:
5299:
5057:
3683:
3168:{\displaystyle H\times A=\left(A,\lbrace e_{i}\mid i\in I_{e},e_{i}\subseteq A\rbrace \right).}
2672:
2594:{\displaystyle H_{A}=\left(A,\lbrace e\cap A\mid e\in E,e\cap A\neq \emptyset \rbrace \right).}
1495:
1460:
10482:
8581:
Huang, Jin; Zhang, Rui; Yu, Jeffrey Xu (2015). "Scalable
Hypergraph Learning and Processing".
5618:
5026:
4995:
4333:
4297:
4231:
1300:: that is, the number of vertices in its tail followed by the number of vertices in its head.
10559:
10427:
8582:
7965:
7246:
5261:
5219:
4974:
3228:
is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by
2376:{\displaystyle E=\lbrace e_{i}\mid i\in I_{e},e_{i}\subseteq X,e_{i}\neq \emptyset \rbrace ,}
2176:
2055:
1569:
Directed hypergraphs can be used to model things including telephony applications, detecting
1543:
1197:
885:
10047:
10008:
9540:
9401:; Pollak, H. O. (2006), "Hypergraph planarity and the complexity of drawing Venn diagrams",
8143:
7591:
As a corollary, an edge-transitive hypergraph that is not vertex-transitive is bicolorable.
10585:
10492:
10462:
10402:
10380:
10277:
9875:
9566:
9045:
8917:
8801:
8747:
8421:
8212:
8002:
7975:
7412:
7136:
7109:
5725:
5473:
4970:
4270:
4073:
4042:
4011:
3919:
3894:
3184:
2645:
2454:
2416:
2389:
1923:
1515:
1480:
1398:
1118:
898:
9159:
7558:
6964:
5936:
5369:
4969:
In contrast with ordinary undirected graphs for which there is a single natural notion of
8:
10502:
10467:
10447:
10345:
8287:"Analyzing Dynamic Hypergraphs with Parallel Aggregated Ordered Hypergraph Visualization"
8233:
8119:
6090:
5996:
1929:
1679:
1621:
1574:
1539:
1445:
1362:
1307:
to a directed hypergraph by defining the head or tail of each edge as a set of vertices (
9436:, Lecture Notes in Computer Science, vol. 5849, Springer-Verlag, pp. 345–356,
9367:, Lecture Notes in Computer Science, vol. 5417, Springer-Verlag, pp. 396–407,
8921:
8805:
8751:
8236: – Abstract mathematical system of two types of objects and a relation between them
5567:
When the edges of a hypergraph are explicitly labeled, one has the additional notion of
10477:
10397:
10118:
9764:
9706:
9544:
9026:
9022:
8941:
8882:
8817:
8771:
8737:
8609:
8544:
8517:
8369:
8333:
8309:
7957:
7598:
7594:
7358:
7282:
6993:
6863:
6843:
6758:
6667:
6647:
6588:
6568:
6112:
6094:
5976:
5956:
5679:
5655:
5598:
5574:
5389:
5154:
3991:
3211:
2953:
2625:
2146:
2082:
2033:
2009:
1987:
1964:
1942:
1882:
1862:
1840:
1647:
1615:
1527:
1174:
1150:
1082:
1054:
986:
966:
930:
9891:
9293:, Lecture Notes in Computer Science, vol. 1984, Springer-Verlag, pp. 45–76,
8788:
Tan, Shulong; Bu, Jiajun; Chen, Chun; Xu, Bin; Wang, Can; He, Xiaofei (October 2011),
10095:
10074:
10053:
10014:
9993:
9910:
9854:
9822:
9789:
9682:
9584:
9552:
9519:
9447:
9378:
9358:
9337:
9304:
9173:
9143:
9126:
8988:
8933:
8904:
Gao, Tue; Wang, Meng; Zha, Zheng-Jun; Shen, Jialie; Li, Xuelong; Wu, Xindong (2013),
8887:
8789:
8763:
8698:
8658:
8599:
8521:
8442:
8337:
8325:
8317:
8286:
7605:
are also important for processing large scale hypergraphs in machine learning tasks.
5170:
1685:
9030:
8983:
8873:
7597:(and in particular, hypergraph partitioning) has many applications to IC design and
7243:
A partition theorem due to E. Dauber states that, for an edge-transitive hypergraph
10407:
10359:
10201:
10196:
9964:
9887:
9842:
9754:
9718:
9616:
9509:
9437:
9412:
9398:
9368:
9329:
9294:
9265:
9222:
9138:
9102:
9054:
9018:
8978:
8945:
8925:
8877:
8869:
8843:
8809:
8790:"Using rich social media information for music recommendation via hypergraph model"
8775:
8755:
8650:
8613:
8591:
8548:
8536:
8480:
8407:
8301:
8134:
7689:
5166:
4682:
3706:
3459:
1573:, operations research, and transportation planning. They can also be used to model
1570:
1519:
9768:
8821:
10089:
10068:
9987:
9783:
9666:
9562:
9442:
9373:
9333:
9058:
8847:
8436:
8417:
8123:
7602:
5644:
When the vertices of a hypergraph are explicitly labeled, one has the notions of
5190:
5162:
5158:
4989:
4985:
4670:
4616:
3437:
1663:
1531:
1453:
444:
8654:
5641:. Note that all strongly isomorphic graphs are isomorphic, but not vice versa.
10390:
10293:
10186:
9620:
8759:
8690:
8474:
8389:
8127:
4226:
1673:
In one possible visual representation for hypergraphs, similar to the standard
1618:- created by augmenting a class of hypergraphs with a set of replacement rules;
1535:
1433:
1304:
9269:
9107:
9090:
8305:
453:
10579:
10298:
10271:
9702:
9523:
9299:
8929:
8321:
8313:
7961:
7704:
5153:
iterative process which removes hyperedges using a generalized definition of
5146:
5142:
1674:
1438:
10073:. Fields Institute Monographs. Vol. 17. American Mathematical Society.
9467:
9007:"A Directed Hypergraph Database: A Model for the Local Loop Telephone Plant"
9006:
8813:
8118:. As this loop is infinitely recursive, sets that are the edges violate the
1701:, for instance, may be viewed as a subdivision drawing of a hypergraph with
888:. In contrast, in an ordinary graph, an edge connects exactly two vertices.
10385:
10265:
9674:
9416:
8992:
8937:
8906:"Visual-textual joint relevance learning for tag-based social image search"
8891:
8767:
8513:
8329:
8218:
7697:
7693:
6074:
5210:
5185:
4981:
1706:
1698:
1551:
1499:
881:
9091:"Directed hypergraphs: Introduction and fundamental algorithms - A survey"
8794:
ACM Transactions on
Multimedia Computing, Communications, and Applications
8649:. Algorithms and Combinatorics. Vol. 29. Springer. pp. 301–317.
4681:
A parallel for the adjacency matrix of a hypergraph can be drawn from the
1783:
colors over all colorings is called the chromatic number of a hypergraph.
10303:
9284:
8595:
8484:
5491:
5364:
5177:
5134:
2451:
is a hypergraph with some vertices removed. Formally, the subhypergraph
2124:. It is generally not reduced, unless all hyperedges have cardinality 1.
1710:
869:
10138:
9759:
9742:
9514:
9497:
8540:
7715:. Consider, for example, the generalized hypergraph whose vertex set is
1589:
and concepts involving graphs also hold for hypergraphs, in particular:
10452:
10364:
10314:
10114:
9934:. Proc. International Conference on Hi Performance Computing (HiPC'95).
9743:"Degrees of Acyclicity for Hypergraphs and Relational Database Schemes"
9498:"Degrees of acyclicity for hypergraphs and relational database schemes"
9227:
8412:
8239:
5145:; it is also equivalent to reducibility to the empty graph through the
1915:
1449:
30:
9968:
9846:
9047:
Exchange
Pattern Mining in the Bitcoin Transaction Directed Hypergraph
8691:"Learning with hypergraphs: clustering, classification, and embedding"
8438:
Heuristics: Intelligent Search Strategies for Computer Problem Solving
6019:. Note that, with this definition of equality, graphs are self-dual:
5189:
to several desirable properties of database schemas and is related to
5133:, then it is Berge-cyclic. Berge-cyclicity can obviously be tested in
2026:
parts, and each hyperedge contains precisely one vertex of each type.
10191:
8393:
8257:
5323:
3889:
2441:
1047:
10110:
9722:
8221: – bipartite graph representing the factorization of a function
7618:
1726:
Classic hypergraph coloring is assigning one of the colors from set
1658:
8374:
8245:
8227:
8206:
7708:
7224:) if all of its vertices are symmetric. Similarly, a hypergraph is
2902:{\displaystyle E'=\lbrace e\in E\mid e\subseteq (A\cup A')\rbrace }
1503:
1105:
8742:
10228:
10135:: open-source PAOHVis system for visualizing dynamic hypergraphs.
4267:
For a directed hypergraph, the heads and tails of each hyperedge
2133:
1586:
9825:
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
9429:
3030:{\displaystyle \left(X,\lbrace e_{i}\mid i\in J\rbrace \right).}
1694:
by coloring, by drawing outlines around them, or both. An order-
9606:"An algorithm for tree-query membership of a distributed query"
9551:, Annals of Discrete Mathematics, vol. 29, North-Holland,
8284:
7684:. In essence, every edge is just an internal node of a tree or
6775:, there does not exist any vertex that meets edges 1, 4 and 6:
3365:{\displaystyle X_{m}=\lbrace e_{i}\mid x_{m}\in e_{i}\rbrace .}
144:{\displaystyle X=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}}
10132:
8476:
Witnesses for non-satisfiability of dense random 3CNF formulas
4980:
A first definition of acyclicity for hypergraphs was given by
1794:. The 2-colorable hypergraphs are exactly the bipartite ones.
10238:
9165:
Proc. 11th International Symposium on Graph Drawing (GD 2003)
8727:
6644:), but they are not strongly isomorphic. So, for example, in
5194:
1448:. In particular, there is a bipartite "incidence graph" or "
9873:
9356:
8689:
Zhou, Dengyong; Huang, Jiayuan; Scholkopf, Bernhard (2006),
7052:
The dual of a uniform hypergraph is regular and vice versa.
1479:(voting games); this notion is applied to solve problems in
10217:
9204:
Eschbach, Thomas; Günther, Wolfgang; Becker, Bernd (2006),
9160:"Layout of directed hypergraphs with orthogonal hyperedges"
9043:
7545:{\displaystyle \sum _{k=1}^{K}r\left(H_{X_{k}}\right)=r(H)}
6745:{\displaystyle e_{1}\cap e_{4}\cap e_{6}=\lbrace a\rbrace }
4949:
4590:
3872:
2950:
of the edge index set, the partial hypergraph generated by
1452:" corresponding to every hypergraph, and conversely, every
9124:
4219:{\displaystyle e_{i}^{*}\in E^{*},~v_{j}^{*}\in e_{i}^{*}}
1786:
Hypergraphs for which there exists a coloring using up to
1580:
10233:
10013:. Lecture Notes in Mathematics. Vol. 411. Springer.
9125:
Gallo, G.; Longo, G.; Pallottino, S.; Nguyen, S. (1993).
8511:
8294:
IEEE Transactions on Visualization and Computer Graphics
2917:
is a hypergraph with some edges removed. Given a subset
9206:"Orthogonal hypergraph drawing for improved visibility"
8262:
Pages displaying short descriptions of redirect targets
2268:{\displaystyle X=\lbrace x_{i}\mid i\in I_{v}\rbrace ,}
1558:
is a hypergraph such that all its hyperedges have size
9646:
9644:
9287:(2001), "Drawing hypergraphs in the subset standard",
9053:. Financial Cryptography and Data Security. Springer.
8242: – Graph with multiple edges between two vertices
8152:
6830:{\displaystyle f_{1}\cap f_{4}\cap f_{6}=\varnothing }
5204:
5193:. Both β-acyclicity and γ-acyclicity can be tested in
4836:
4439:
3785:
9947:
IEEE Transactions on Parallel and Distributed Systems
9203:
8146:
8078:
8032:
8005:
7978:
7960:
of set membership for such hypergraphs does induce a
7929:
7889:
7856:
7804:
7759:
7721:
7561:
7477:
7442:
7415:
7381:
7361:
7293:
7249:
7170:
7139:
7112:
7077:
6996:
6967:
6912:
6886:
6866:
6846:
6784:
6761:
6693:
6670:
6650:
6611:
6591:
6571:
6356:
6138:
6115:
6028:
5999:
5979:
5959:
5939:
5896:
5870:
5806:
5751:
5728:
5702:
5682:
5658:
5621:
5601:
5577:
5529:
5503:
5476:
5415:
5392:
5372:
5334:
5302:
5264:
5222:
5097:
5060:
5029:
4998:
4814:
4770:
4737:
4695:
4417:
4372:
4336:
4300:
4273:
4234:
4148:
4103:
4076:
4045:
4014:
3994:
3928:
3897:
3763:
3718:
3686:
3612:
3535:
3388:
3303:
3267:
3234:
3214:
3187:
3079:
3046:
2979:
2956:
2923:
2843:
2790:
2717:
2675:
2648:
2628:
2513:
2484:
2457:
2419:
2392:
2290:
2220:
2179:
2085:
2058:
2036:
2012:
1990:
1967:
1945:
1885:
1865:
1843:
1732:
1463:, an undirected hypergraph may sometimes be called a
1401:
1365:
1339:
1313:
1238:
1200:
1177:
1153:
1121:
1085:
1057:
1009:
989:
969:
933:
901:
817:
759:
713:
667:
618:
525:
463:
410:
349:
301:
237:
157:
40:
9468:"Vitaly Voloshin: Mixed Hypergraph Coloring Website"
8223:
Pages displaying wikidata descriptions as a fallback
5615:
if the permutation is the identity. One then writes
3474:
is a host graph if there is a bijection between the
606:{\displaystyle E=\{a_{1},a_{2},a_{3},a_{4},a_{5}\}=}
9736:
9734:
9732:
9651:Graham, M. H. (1979). "On the universal relation".
9641:
9323:
9256:Mäkinen, Erkki (1990), "How to draw a hypergraph",
8368:(2020), "Hypergraphs: an introduction and review",
7700:and vertices correspond to constants or variables.
4648:) are connected with an edge if and only if vertex
3443:with the same vertex set as a connected hypergraph
1905:- its vertices can be partitioned into two classes
1806:A hypergraph can have various properties, such as:
10006:
9876:"Graph partitioning models for parallel computing"
9665:
9655:. Toronto, Ontario, Canada: University of Toronto.
8688:
8188:
8110:
8064:
8018:
7991:
7948:
7915:
7875:
7842:
7790:
7745:
7576:
7544:
7460:
7428:
7401:
7367:
7344:
7273:
7205:
7152:
7125:
7098:
7002:
6982:
6938:
6898:
6872:
6852:
6829:
6767:
6744:
6676:
6656:
6632:
6597:
6577:
6554:
6336:
6121:
6062:
6011:
5985:
5965:
5945:
5922:
5882:
5850:
5786:
5734:
5714:
5688:
5664:
5633:
5607:
5583:
5555:
5515:
5482:
5459:
5398:
5378:
5352:
5314:
5288:
5246:
5125:
5083:
5046:
5015:
4954:
4798:
4756:
4723:
4595:
4400:
4358:
4322:
4286:
4256:
4218:
4134:
4089:
4058:
4027:
4000:
3976:
3910:
3877:
3746:
3698:
3672:
3598:
3425:
3364:
3286:
3253:
3220:
3200:
3167:
3058:
3029:
2962:
2942:
2901:
2829:
2776:
2700:
2661:
2642:which is partially contained in the subhypergraph
2634:
2593:
2496:
2470:
2432:
2405:
2375:
2267:
2203:
2091:
2070:
2042:
2018:
1996:
1973:
1951:
1891:
1871:
1849:
1774:
1419:
1381:
1351:
1325:
1292:
1224:
1183:
1159:
1139:
1091:
1063:
1033:
995:
975:
939:
919:
856:
804:
746:
700:
654:
605:
511:
432:
397:
336:
288:
224:
143:
9944:
9929:
8695:Advances in Neural Information Processing Systems
8640:
8584:2015 IEEE International Conference on Data Mining
8522:"On the Desirability of Acyclic Database Schemes"
8473:Feige, Uriel; Kim, Jeong Han; Ofek, Eran (2006).
8441:. Addison-Wesley Publishing Company. p. 25.
4731:for a hypergraph in general where the hyperedges
3599:{\displaystyle V=\{v_{1},v_{2},~\ldots ,~v_{n}\}}
10577:
10119:Creative Commons Attribution/Share-Alike License
9729:
9701:
3673:{\displaystyle E=\{e_{1},e_{2},~\ldots ~e_{m}\}}
2830:{\displaystyle A'=\bigcup _{e\in E}e\setminus A}
10010:Hypergraph Seminar: Ohio State University, 1972
9361:(2009), "Subdivision drawings of hypergraphs",
8859:
8833:
4689:. Likewise, we can define the adjacency matrix
1483:. In some literature edges are referred to as
8968:
8903:
8230: – Set system used in greedy optimization
225:{\displaystyle E=\{e_{1},e_{2},e_{3},e_{4}\}=}
10330:
10154:
9539:
9397:
9282:
9258:International Journal of Computer Mathematics
9088:
8388:
8260: – Model to describe distributed systems
34:An example of an undirected hypergraph, with
9915:: CS1 maint: multiple names: authors list (
9859:: CS1 maint: multiple names: authors list (
9214:Journal of Graph Algorithms and Applications
9172:, vol. 2912, Springer, pp. 381–6,
8580:
8472:
8396:(1987), "ε-nets and simplex range queries",
8215: – Symmetric arrangement of finite sets
8105:
8092:
8059:
8046:
7837:
7818:
7785:
7773:
7740:
7728:
6739:
6733:
6549:
6546:
6534:
6515:
6503:
6484:
6472:
6453:
6441:
6422:
6410:
6391:
6379:
6363:
6331:
6328:
6316:
6297:
6285:
6266:
6254:
6235:
6223:
6204:
6192:
6173:
6161:
6145:
3667:
3619:
3593:
3542:
3356:
3317:
3281:
3268:
3248:
3235:
3154:
3103:
3016:
2991:
2896:
2855:
2580:
2538:
2367:
2297:
2259:
2227:
1801:
1769:
1733:
851:
845:
839:
833:
821:
793:
781:
775:
763:
735:
729:
723:
717:
689:
683:
677:
671:
643:
637:
631:
625:
619:
597:
532:
506:
470:
427:
424:
411:
389:
350:
328:
302:
280:
241:
238:
216:
164:
138:
47:
10091:Introduction to Graph and Hypergraph Theory
9603:
9084:
9082:
8787:
7647:. Unsourced material may be challenged and
7345:{\displaystyle (X_{1},X_{2},\cdots ,X_{K})}
7026:. A graph is just a 2-uniform hypergraph.
10337:
10323:
10161:
10147:
8647:Optimal Interconnection Trees in the Plane
8507:
8505:
7608:
7164:if there exists an automorphism such that
7071:if there exists an automorphism such that
5137:by an exploration of the incidence graph.
1678:curves that connect sets of points, or as
1546:with hypergraph Laplacian, and hypergraph
457:An example of a directed hypergraph, with
10168:
9989:Hypergraphs: Combinatorics of Finite Sets
9958:
9836:
9758:
9513:
9441:
9372:
9298:
9226:
9142:
9106:
9004:
8982:
8881:
8741:
8643:"Steiner Trees in Graphs and Hypergraphs"
8411:
8373:
7667:Learn how and when to remove this message
6906:, and the duals are strongly isomorphic:
4799:{\displaystyle w_{e_{k}}\in \mathbb {R} }
4792:
3426:{\displaystyle \left(H^{*}\right)^{*}=H.}
2211:be the hypergraph consisting of vertices
2100:-uniform and bipartite (and 2-colorable).
10344:
10109:This article incorporates material from
10087:
10066:
9326:Database and Expert Systems Applications
9120:
9118:
9089:Ausiello, Giorgio; Laura, Luigi (2017).
9079:
8576:
8574:
8572:
7037:is the number of edges that contain it.
6063:{\displaystyle \left(H^{*}\right)^{*}=H}
5851:{\displaystyle \phi (e_{i})=f_{\pi (i)}}
5460:{\displaystyle \phi (e_{i})=f_{\pi (i)}}
2622:is a hypergraph where each hyperedge of
2444:of the vertices and edges respectively.
2127:An abstract simplicial complex with the
1684:
1657:
1303:The definition above generalizes from a
452:
443:
289:{\displaystyle \{\{v_{1},v_{2},v_{3}\},}
29:
9255:
9127:"Directed hypergraphs and applications"
8502:
8364:
7964:, and "flattens" the hypergraph into a
4984:: a hypergraph is Berge-acyclic if its
2777:{\displaystyle Ex(H_{A})=(A\cup A',E')}
1581:Generalizations of concepts from graphs
1522:tasks as the data model and classifier
1459:Hypergraphs have many other names. In
14:
10578:
10045:
9930:Catalyurek, U.V.; Aykanat, C. (1995).
9781:
9650:
9357:Kaufmann, Michael; van Kreveld, Marc;
9157:
8280:
8278:
5149:(also known as Graham's algorithm), a
2153:
1775:{\displaystyle \{1,2,3,...,\lambda \}}
1721:
1389:will generalize to hypergraph theory.
398:{\displaystyle \{v_{3},v_{5},v_{6}\},}
10318:
10142:
10007:Berge, C.; Ray-Chaudhuri, D. (2006).
9985:
9874:Hendrickson, B., Kolda, T.G. (2000),
9740:
9578:
9535:
9533:
9495:
9115:
8910:IEEE Transactions on Image Processing
8569:
8434:
3977:{\displaystyle H^{*}=(V^{*},\ E^{*})}
3486:, such that each connected component
3287:{\displaystyle \lbrace X_{m}\rbrace }
3254:{\displaystyle \lbrace e_{i}\rbrace }
1834:- has no loops and no repeated edges.
1653:
1518:. They have been extensively used in
9604:Yu, C. T.; Özsoyoğlu, M. Z. (1979).
7645:adding citations to reliable sources
7612:
2669:is fully contained in the extension
2006:- the vertices are partitioned into
1961:- each hyperedge contains precisely
8399:Discrete and Computational Geometry
8275:
8252:Sparse matrix–vector multiplication
8209: – Type of directed hypergraph
5205:Isomorphism, symmetry, and equality
4676:
3524:
1494:The collection of hypergraphs is a
1467:and then the hyperedges are called
24:
10049:Hypergraph Theory: An Introduction
9530:
9023:10.1002/j.1538-7305.1982.tb03439.x
8641:Brazil, M; Zachariasen, M (2015).
8126:, but is rather just some general
7603:hypergraph partitioning algorithms
7206:{\displaystyle \phi (e_{i})=e_{j}}
6097:of the hypergraph and written Aut(
5787:{\displaystyle \phi (x_{n})=y_{n}}
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4862:
4859:
4606:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4506:
4503:
4454:
4451:
4135:{\displaystyle v_{j}^{*}\in V^{*}}
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3797:
3794:
2577:
2364:
1640:Hall-type theorems for hypergraphs
1534:(correlations as hyperedges), and
805:{\displaystyle (\{2,3\},\{4,5\}),}
25:
10607:
10126:
9170:Lecture Notes in Computer Science
7843:{\displaystyle e_{2}=\{a,e_{1}\}}
6824:
6684:meets edges 1, 4 and 6, so that,
5933:If, in addition, the permutation
5556:{\displaystyle H^{*}\simeq G^{*}}
2821:
857:{\displaystyle (\{3,5\},\{6\})\}}
512:{\displaystyle X=\{1,2,3,4,5,6\}}
7617:
6939:{\displaystyle H^{*}\cong G^{*}}
6633:{\displaystyle \phi (a)=\alpha }
5923:{\displaystyle H^{*}\cong G^{*}}
3466:. For a disconnected hypergraph
1718:for simple hypergraphs as well.
983:is a set of pairs of subsets of
655:{\displaystyle \{(\{1\},\{2\}),}
337:{\displaystyle \{v_{2},v_{3}\},}
9938:
9923:
9898:from the original on 2021-01-26
9867:
9816:
9802:from the original on 2023-02-04
9775:
9695:
9659:
9630:from the original on 2018-09-02
9597:
9572:
9489:
9478:from the original on 2022-01-20
9460:
9423:
9391:
9350:
9317:
9276:
9249:
9238:from the original on 2011-07-18
9197:
9186:from the original on 2011-07-18
9151:
9068:from the original on 2021-07-15
9037:
8998:
8962:
8952:from the original on 2017-09-23
8897:
8853:
8827:
8781:
8721:
8711:from the original on 2021-10-22
8671:from the original on 2021-01-29
8623:from the original on 2021-01-26
8558:from the original on 2021-04-21
8491:from the original on 2021-01-27
8455:from the original on 2023-02-04
8347:from the original on 2021-01-26
8111:{\displaystyle e_{2}=\{e_{1}\}}
8065:{\displaystyle e_{1}=\{e_{2}\}}
7014:, the hypergraph is said to be
5953:is the identity, one says that
3494:is a host of the corresponding
1509:
1475:theory, hypergraphs are called
10488:Cremona–Richmond configuration
10117:, which is licensed under the
8697:, MIT Press, pp. 1601–8,
8682:
8634:
8466:
8428:
8382:
8358:
7916:{\displaystyle e_{1}\in e_{2}}
7571:
7565:
7539:
7533:
7339:
7294:
7268:
7256:
7187:
7174:
7087:
7081:
6977:
6971:
6621:
6615:
6093:under composition, called the
5843:
5837:
5823:
5810:
5768:
5755:
5452:
5446:
5432:
5419:
5344:
5283:
5271:
5241:
5229:
5180:if a hypergraph is α-acyclic.
4895:
4869:
4718:
4702:
4542:
4529:
4490:
4477:
4395:
4379:
4353:
4340:
4317:
4304:
3971:
3942:
3741:
3725:
3712:For an undirected hypergraph,
2943:{\displaystyle J\subset I_{e}}
2893:
2876:
2771:
2743:
2737:
2724:
2695:
2682:
2198:
2186:
1650:, and shortest path problems.
1530:(communities as hyperedges),
1414:
1402:
1375:
1367:
1287:
1283:
1275:
1267:
1259:
1255:
1248:
1240:
1219:
1207:
1134:
1122:
1022:
1010:
914:
902:
848:
818:
796:
760:
747:{\displaystyle (\{3\},\{1\}),}
738:
714:
701:{\displaystyle (\{2\},\{3\}),}
692:
668:
646:
622:
27:Generalization of graph theory
13:
1:
9979:
9892:10.1016/S0167-8191(00)00048-X
9011:Bell System Technical Journal
8984:10.1093/bioinformatics/btp467
8874:10.1093/bioinformatics/btt224
8026:, and zero vertices, so that
7791:{\displaystyle e_{1}=\{a,b\}}
7461:{\displaystyle 1\leq k\leq K}
7238:
3261:and whose edges are given by
1879:, i.e., contained in exactly
1682:that enclose sets of points.
1444:Hypergraphs can be viewed as
1293:{\displaystyle |e|=(|D|,|C|)}
1147:is the number of vertices in
10565:Kirkman's schoolgirl problem
10498:Grünbaum–Rigby configuration
10268:for cubic Hamiltonian graphs
10088:Voloshin, Vitaly I. (2009).
10067:Voloshin, Vitaly I. (2002).
9583:. Amsterdam: North-Holland.
9496:Fagin, Ronald (1983-07-01).
9443:10.1007/978-3-642-11805-0_33
9374:10.1007/978-3-642-00219-9_39
9334:10.1007/978-3-319-64471-4_31
9144:10.1016/0166-218X(93)90045-P
9131:Discrete Applied Mathematics
9095:Theoretical Computer Science
9059:10.1007/978-3-319-70278-0_16
8848:10.1016/j.patcog.2010.07.014
7375:such that the subhypergraph
5353:{\displaystyle \phi :X\to Y}
3922:matrix defines a hypergraph
3059:{\displaystyle A\subseteq X}
2620:extension of a subhypergraph
2497:{\displaystyle A\subseteq X}
1922:Two stronger properties are
1524:regularization (mathematics)
1352:{\displaystyle D\subseteq X}
1326:{\displaystyle C\subseteq X}
1232:in a directed hypergraph is
947:is a set of elements called
7:
10458:Möbius–Kantor configuration
10034:Encyclopedia of Mathematics
8655:10.1007/978-3-319-13915-9_5
8254: – Computation routine
8248: – Computational model
8199:
7216:A hypergraph is said to be
7045:if every vertex has degree
6949:
6104:
4757:{\displaystyle e_{k\leq m}}
4070:-element set of subsets of
2604:An alternative term is the
2121:abstract simplicial complex
1600:Vertex cover in hypergraphs
1526:. The applications include
433:{\displaystyle \{v_{4}\}\}}
10:
10612:
10544:Bruck–Ryser–Chowla theorem
10182:Graph (abstract data type)
9788:. CRC Press. p. 172.
9621:10.1109/CMPSAC.1979.762509
8760:10.1103/PhysRevE.79.066118
7949:{\displaystyle b\in e_{2}}
7876:{\displaystyle b\in e_{1}}
7099:{\displaystyle \phi (x)=y}
5494:of the graphs. Note that
5126:{\displaystyle v,v'\in f'}
5023:of vertices and some pair
4724:{\displaystyle A=(a_{ij})}
4401:{\displaystyle I=(b_{ij})}
3747:{\displaystyle I=(b_{ij})}
3680:. Every hypergraph has an
3070:is the partial hypergraph
1859:- every vertex has degree
1790:colors are referred to as
1610:Line graph of a hypergraph
1171:is the number of edges in
1034:{\displaystyle (D,C)\in E}
10552:
10534:Szemerédi–Trotter theorem
10516:
10438:
10373:
10352:
10286:
10247:
10210:
10174:
9711:SIAM Journal on Computing
9270:10.1080/00207169008803875
9108:10.1016/j.tcs.2016.03.016
8306:10.1109/TVCG.2019.2933196
7746:{\displaystyle V=\{a,b\}}
7601:. Efficient and scalable
7402:{\displaystyle H_{X_{k}}}
6899:{\displaystyle H\equiv G}
5883:{\displaystyle H\equiv G}
5715:{\displaystyle H\equiv G}
5516:{\displaystyle H\simeq G}
5315:{\displaystyle H\simeq G}
5084:{\displaystyle v,v'\in f}
4964:
3699:{\displaystyle n\times m}
3462:a connected subgraph in
2701:{\displaystyle Ex(H_{A})}
1802:Properties of hypergraphs
1646:In directed hypergraphs:
876:is a generalization of a
10524:Sylvester–Gallai theorem
10260:Graph Modelling Language
9882:(Submitted manuscript),
9679:Foundations of Databases
9300:10.1007/3-540-44541-2_15
8930:10.1109/tip.2012.2202676
8268:
6109:Consider the hypergraph
5634:{\displaystyle H\cong G}
5054:of hyperedges such that
5047:{\displaystyle f\neq f'}
5016:{\displaystyle v\neq v'}
4615:may be represented by a
4359:{\displaystyle T(e_{j})}
4323:{\displaystyle H(e_{j})}
4257:{\displaystyle b_{ij}=1}
1548:semi-supervised learning
10596:Mathematical structures
10529:De Bruijn–Erdős theorem
10473:Desargues configuration
9404:Journal of Graph Theory
8814:10.1145/2037676.2037679
7609:Further generalizations
7436:is transitive for each
7274:{\displaystyle H=(X,E)}
5289:{\displaystyle G=(Y,F)}
5247:{\displaystyle H=(X,E)}
2204:{\displaystyle H=(X,E)}
2071:{\displaystyle k\geq 2}
1636:on uniform hypergraphs;
1594:Matching in hypergraphs
1225:{\displaystyle e=(D,C)}
884:can join any number of
10046:Bretto, Alain (2013).
9986:Berge, Claude (1984).
9741:Fagin, Ronald (1983).
9581:Graphs and Hypergraphs
9579:Berge, Claude (1973).
9417:10.1002/jgt.3190110306
9005:Goldstein, A. (1982).
8190:
8189:{\displaystyle \left.}
8112:
8066:
8020:
7993:
7950:
7923:, it is not true that
7917:
7877:
7844:
7792:
7747:
7713:directed acyclic graph
7696:; edges correspond to
7686:directed acyclic graph
7578:
7546:
7498:
7462:
7430:
7403:
7369:
7346:
7275:
7207:
7154:
7127:
7100:
7004:
6984:
6940:
6900:
6874:
6854:
6831:
6769:
6746:
6678:
6658:
6634:
6599:
6579:
6556:
6338:
6123:
6064:
6013:
5987:
5967:
5947:
5924:
5884:
5852:
5788:
5736:
5716:
5690:
5666:
5635:
5609:
5585:
5557:
5517:
5484:
5461:
5400:
5380:
5354:
5316:
5290:
5248:
5127:
5085:
5048:
5017:
4956:
4800:
4758:
4725:
4597:
4402:
4360:
4324:
4288:
4258:
4220:
4136:
4091:
4060:
4029:
4002:
3978:
3912:
3879:
3748:
3700:
3674:
3600:
3455:if every hyperedge of
3427:
3366:
3288:
3255:
3222:
3202:
3169:
3060:
3031:
2964:
2944:
2903:
2831:
2778:
2702:
2663:
2636:
2595:
2498:
2472:
2434:
2407:
2377:
2269:
2205:
2093:
2072:
2044:
2020:
1998:
1975:
1953:
1893:
1873:
1851:
1776:
1690:
1667:
1634:Kruskal–Katona theorem
1461:computational geometry
1421:
1383:
1353:
1327:
1294:
1226:
1185:
1169:size of the hypergraph
1161:
1141:
1093:
1065:
1035:
1003:. Each of these pairs
997:
977:
941:
921:
865:
858:
806:
748:
702:
656:
607:
513:
450:
441:
434:
399:
338:
290:
226:
145:
10560:Design of experiments
10169:Graph representations
8435:Pearl, Judea (1984).
8191:
8113:
8067:
8021:
8019:{\displaystyle e_{2}}
7994:
7992:{\displaystyle e_{1}}
7966:partially ordered set
7951:
7918:
7878:
7845:
7793:
7748:
7579:
7547:
7478:
7463:
7431:
7429:{\displaystyle X_{k}}
7404:
7370:
7347:
7276:
7208:
7155:
7153:{\displaystyle e_{j}}
7128:
7126:{\displaystyle e_{i}}
7101:
7005:
6985:
6941:
6901:
6875:
6855:
6832:
6770:
6747:
6679:
6659:
6635:
6605:are isomorphic (with
6600:
6580:
6557:
6339:
6124:
6065:
6014:
5988:
5968:
5948:
5925:
5885:
5853:
5789:
5737:
5735:{\displaystyle \phi }
5717:
5691:
5667:
5636:
5610:
5586:
5558:
5518:
5485:
5483:{\displaystyle \phi }
5462:
5401:
5381:
5355:
5317:
5291:
5249:
5161:, it is known that a
5128:
5086:
5049:
5018:
4957:
4801:
4759:
4726:
4655:is contained in edge
4622:as follows: the sets
4598:
4403:
4361:
4325:
4289:
4287:{\displaystyle e_{j}}
4259:
4221:
4137:
4092:
4090:{\displaystyle V^{*}}
4061:
4059:{\displaystyle E^{*}}
4030:
4028:{\displaystyle V^{*}}
4003:
3979:
3913:
3911:{\displaystyle I^{t}}
3880:
3749:
3701:
3675:
3601:
3428:
3367:
3289:
3256:
3223:
3203:
3201:{\displaystyle H^{*}}
3170:
3061:
3032:
2965:
2945:
2904:
2832:
2779:
2703:
2664:
2662:{\displaystyle H_{A}}
2637:
2596:
2499:
2473:
2471:{\displaystyle H_{A}}
2435:
2433:{\displaystyle I_{e}}
2408:
2406:{\displaystyle I_{v}}
2378:
2270:
2206:
2129:augmentation property
2094:
2073:
2045:
2021:
1999:
1976:
1954:
1894:
1874:
1852:
1777:
1688:
1661:
1628:Erdős–Ko–Rado theorem
1544:spectral graph theory
1516:Steiner tree problems
1422:
1420:{\displaystyle (X,E)}
1394:undirected hypergraph
1384:
1354:
1328:
1295:
1227:
1186:
1162:
1142:
1140:{\displaystyle (X,E)}
1114:order of a hypergraph
1094:
1066:
1051:; the vertex subset
1036:
998:
978:
942:
922:
920:{\displaystyle (X,E)}
859:
807:
749:
703:
657:
608:
514:
456:
447:
435:
400:
339:
291:
227:
146:
33:
10493:Kummer configuration
10463:Pappus configuration
10346:Incidence structures
10278:Trivial Graph Format
9782:Harary, F. (2018) .
9283:Bertault, François;
8842:(10–11): 2255–2262,
8596:10.1109/ICDM.2015.33
8590:. pp. 775–780.
8485:10.1109/FOCS.2006.78
8213:Combinatorial design
8144:
8076:
8030:
8003:
7976:
7927:
7887:
7854:
7802:
7757:
7753:and whose edges are
7719:
7641:improve this section
7577:{\displaystyle r(H)}
7559:
7475:
7440:
7413:
7379:
7359:
7291:
7247:
7168:
7137:
7110:
7075:
6994:
6983:{\displaystyle r(H)}
6965:
6910:
6884:
6864:
6844:
6782:
6759:
6691:
6668:
6648:
6609:
6589:
6569:
6354:
6136:
6113:
6026:
5997:
5977:
5957:
5946:{\displaystyle \pi }
5937:
5894:
5868:
5804:
5749:
5726:
5700:
5680:
5656:
5619:
5599:
5575:
5527:
5501:
5474:
5413:
5390:
5379:{\displaystyle \pi }
5370:
5332:
5300:
5262:
5220:
5095:
5058:
5027:
4996:
4812:
4768:
4735:
4693:
4415:
4370:
4334:
4298:
4271:
4232:
4146:
4101:
4074:
4043:
4012:
3992:
3926:
3895:
3761:
3716:
3684:
3610:
3533:
3476:connected components
3386:
3301:
3265:
3232:
3212:
3185:
3077:
3044:
2977:
2954:
2921:
2841:
2788:
2715:
2673:
2646:
2626:
2511:
2482:
2455:
2417:
2390:
2288:
2218:
2177:
2083:
2056:
2034:
2010:
1988:
1965:
1943:
1883:
1863:
1841:
1730:
1680:simple closed curves
1556:k-uniform hypergraph
1481:social choice theory
1446:incidence structures
1399:
1363:
1337:
1311:
1236:
1198:
1175:
1151:
1119:
1083:
1055:
1007:
987:
967:
931:
899:
815:
757:
711:
665:
616:
523:
461:
408:
347:
299:
235:
155:
38:
10503:Klein configuration
10483:Schläfli double six
10468:Hesse configuration
10448:Complete quadrangle
9760:10.1145/2402.322390
9515:10.1145/2402.322390
9158:Sander, G. (2003),
8922:2013ITIP...22..363Y
8836:Pattern Recognition
8806:2011smma.book..213T
8752:2009PhRvE..79f6118G
8541:10.1145/2402.322389
8234:Incidence structure
8120:axiom of foundation
6012:{\displaystyle H=G}
5722:if the isomorphism
5593:strongly isomorphic
5490:is then called the
5157:. In the domain of
4215:
4197:
4163:
4118:
2168:section hypergraphs
2164:partial hypergraphs
2154:Related hypergraphs
1722:Hypergraph coloring
1575:Horn-satisfiability
1540:spectral clustering
1382:{\displaystyle |e|}
893:directed hypergraph
10478:Reye configuration
10248:Text-based formats
9880:Parallel Computing
9747:Journal of the ACM
9681:. Addison-Wesley.
9613:Proc. IEEE COMPSAC
9502:Journal of the ACM
9359:Speckmann, Bettina
9228:10.7155/jgaa.00122
8868:(10–11): 237–246,
8529:Journal of the ACM
8413:10.1007/BF02187876
8186:
8177:
8108:
8062:
8016:
7989:
7958:transitive closure
7946:
7913:
7873:
7840:
7788:
7743:
7599:parallel computing
7595:Graph partitioning
7574:
7542:
7458:
7426:
7399:
7365:
7355:of the vertex set
7342:
7271:
7203:
7150:
7123:
7096:
7000:
6980:
6936:
6896:
6870:
6850:
6827:
6765:
6742:
6674:
6654:
6630:
6595:
6575:
6552:
6334:
6119:
6095:automorphism group
6060:
6009:
5983:
5963:
5943:
5920:
5880:
5848:
5784:
5732:
5712:
5686:
5662:
5631:
5605:
5581:
5569:strong isomorphism
5553:
5513:
5480:
5457:
5396:
5376:
5350:
5322:if there exists a
5312:
5286:
5244:
5123:
5081:
5044:
5013:
4952:
4947:
4796:
4764:have real weights
4754:
4721:
4593:
4588:
4398:
4356:
4320:
4284:
4254:
4216:
4201:
4183:
4149:
4132:
4104:
4087:
4056:
4025:
3998:
3974:
3908:
3875:
3870:
3744:
3696:
3670:
3596:
3511:representing graph
3423:
3362:
3284:
3251:
3218:
3198:
3165:
3068:section hypergraph
3056:
3027:
2970:is the hypergraph
2960:
2940:
2915:partial hypergraph
2899:
2827:
2817:
2774:
2698:
2659:
2632:
2591:
2494:
2468:
2430:
2403:
2373:
2265:
2201:
2147:laminar set family
2089:
2068:
2040:
2016:
1994:
1971:
1949:
1889:
1869:
1847:
1772:
1691:
1668:
1654:Hypergraph drawing
1648:transitive closure
1616:Hypergraph grammar
1528:recommender system
1417:
1379:
1349:
1323:
1290:
1222:
1181:
1157:
1137:
1089:
1061:
1031:
993:
973:
937:
917:
866:
854:
802:
744:
698:
652:
603:
509:
451:
442:
430:
395:
334:
286:
222:
141:
10573:
10572:
10312:
10311:
10211:XML-based formats
10101:978-1-61470-112-5
10080:978-0-8218-2812-0
10059:978-3-319-00080-0
10020:978-3-540-37803-7
9999:978-0-08-088023-5
9969:10.1109/71.780863
9886:(12): 1519–1545,
9847:10.1109/92.748202
9795:978-0-429-96231-8
9472:spectrum.troy.edu
9453:978-3-642-11804-3
9399:Johnson, David S.
9384:978-3-642-00218-2
9343:978-3-319-64470-7
9310:978-3-540-41554-1
9179:978-3-540-24595-7
8977:(21): 2831–2838,
8800:(1), Article 22,
8730:Physical Review E
8664:978-3-319-13915-9
8605:978-1-4673-9504-5
8448:978-0-201-05594-8
7850:. Then, although
7677:
7676:
7669:
7368:{\displaystyle X}
7281:, there exists a
7218:vertex-transitive
7022:, or is called a
7003:{\displaystyle H}
6873:{\displaystyle G}
6853:{\displaystyle H}
6840:In this example,
6768:{\displaystyle G}
6677:{\displaystyle a}
6657:{\displaystyle H}
6598:{\displaystyle G}
6578:{\displaystyle H}
6122:{\displaystyle H}
5986:{\displaystyle G}
5966:{\displaystyle H}
5689:{\displaystyle G}
5665:{\displaystyle H}
5608:{\displaystyle G}
5584:{\displaystyle H}
5399:{\displaystyle I}
5171:first-order logic
4868:
4630:are the parts of
4512:
4460:
4182:
4039:-element set and
4001:{\displaystyle H}
3960:
3803:
3656:
3650:
3582:
3573:
3221:{\displaystyle H}
2963:{\displaystyle J}
2802:
2635:{\displaystyle H}
2092:{\displaystyle k}
2052:hypergraph (for
2043:{\displaystyle k}
2019:{\displaystyle k}
1997:{\displaystyle k}
1974:{\displaystyle k}
1952:{\displaystyle k}
1892:{\displaystyle d}
1872:{\displaystyle d}
1850:{\displaystyle d}
1542:that extends the
1184:{\displaystyle E}
1160:{\displaystyle X}
1092:{\displaystyle C}
1064:{\displaystyle D}
996:{\displaystyle X}
976:{\displaystyle E}
940:{\displaystyle X}
16:(Redirected from
10603:
10591:Families of sets
10408:Projective plane
10360:Incidence matrix
10339:
10332:
10325:
10316:
10315:
10287:Related concepts
10202:Incidence matrix
10197:Adjacency matrix
10163:
10156:
10149:
10140:
10139:
10105:
10094:. Nova Science.
10084:
10063:
10042:
10024:
10003:
9973:
9972:
9962:
9942:
9936:
9935:
9927:
9921:
9920:
9914:
9906:
9904:
9903:
9871:
9865:
9864:
9858:
9850:
9840:
9820:
9814:
9813:
9808:
9807:
9779:
9773:
9772:
9762:
9738:
9727:
9726:
9699:
9693:
9692:
9663:
9657:
9656:
9653:Technical Report
9648:
9639:
9638:
9636:
9635:
9629:
9610:
9601:
9595:
9594:
9576:
9570:
9569:
9537:
9528:
9527:
9517:
9493:
9487:
9486:
9484:
9483:
9464:
9458:
9456:
9445:
9427:
9421:
9419:
9395:
9389:
9387:
9376:
9354:
9348:
9346:
9321:
9315:
9313:
9302:
9280:
9274:
9272:
9253:
9247:
9245:
9244:
9243:
9237:
9230:
9210:
9201:
9195:
9193:
9192:
9191:
9155:
9149:
9148:
9146:
9137:(2–3): 177–201.
9122:
9113:
9112:
9110:
9086:
9077:
9076:
9074:
9073:
9067:
9052:
9041:
9035:
9034:
9002:
8996:
8995:
8986:
8966:
8960:
8959:
8958:
8957:
8901:
8895:
8894:
8885:
8857:
8851:
8850:
8831:
8825:
8824:
8785:
8779:
8778:
8745:
8725:
8719:
8718:
8717:
8716:
8686:
8680:
8679:
8677:
8676:
8638:
8632:
8631:
8629:
8628:
8622:
8589:
8578:
8567:
8566:
8564:
8563:
8557:
8526:
8509:
8500:
8499:
8497:
8496:
8470:
8464:
8463:
8461:
8460:
8432:
8426:
8424:
8415:
8386:
8380:
8378:
8377:
8362:
8356:
8355:
8353:
8352:
8346:
8291:
8282:
8263:
8224:
8195:
8193:
8192:
8187:
8182:
8178:
8135:incidence matrix
8117:
8115:
8114:
8109:
8104:
8103:
8088:
8087:
8071:
8069:
8068:
8063:
8058:
8057:
8042:
8041:
8025:
8023:
8022:
8017:
8015:
8014:
7998:
7996:
7995:
7990:
7988:
7987:
7956:. However, the
7955:
7953:
7952:
7947:
7945:
7944:
7922:
7920:
7919:
7914:
7912:
7911:
7899:
7898:
7882:
7880:
7879:
7874:
7872:
7871:
7849:
7847:
7846:
7841:
7836:
7835:
7814:
7813:
7797:
7795:
7794:
7789:
7769:
7768:
7752:
7750:
7749:
7744:
7690:computer science
7672:
7665:
7661:
7658:
7652:
7621:
7613:
7583:
7581:
7580:
7575:
7551:
7549:
7548:
7543:
7526:
7522:
7521:
7520:
7519:
7497:
7492:
7468:, and such that
7467:
7465:
7464:
7459:
7435:
7433:
7432:
7427:
7425:
7424:
7408:
7406:
7405:
7400:
7398:
7397:
7396:
7395:
7374:
7372:
7371:
7366:
7351:
7349:
7348:
7343:
7338:
7337:
7319:
7318:
7306:
7305:
7280:
7278:
7277:
7272:
7222:vertex-symmetric
7212:
7210:
7209:
7204:
7202:
7201:
7186:
7185:
7160:are said to be
7159:
7157:
7156:
7151:
7149:
7148:
7132:
7130:
7129:
7124:
7122:
7121:
7105:
7103:
7102:
7097:
7009:
7007:
7006:
7001:
6990:of a hypergraph
6989:
6987:
6986:
6981:
6960:
6959:
6945:
6943:
6942:
6937:
6935:
6934:
6922:
6921:
6905:
6903:
6902:
6897:
6880:are equivalent,
6879:
6877:
6876:
6871:
6859:
6857:
6856:
6851:
6836:
6834:
6833:
6828:
6820:
6819:
6807:
6806:
6794:
6793:
6774:
6772:
6771:
6766:
6751:
6749:
6748:
6743:
6729:
6728:
6716:
6715:
6703:
6702:
6683:
6681:
6680:
6675:
6663:
6661:
6660:
6655:
6639:
6637:
6636:
6631:
6604:
6602:
6601:
6596:
6584:
6582:
6581:
6576:
6561:
6559:
6558:
6553:
6530:
6529:
6499:
6498:
6468:
6467:
6437:
6436:
6406:
6405:
6375:
6374:
6343:
6341:
6340:
6335:
6312:
6311:
6281:
6280:
6250:
6249:
6219:
6218:
6188:
6187:
6157:
6156:
6128:
6126:
6125:
6120:
6069:
6067:
6066:
6061:
6053:
6052:
6047:
6043:
6042:
6018:
6016:
6015:
6010:
5992:
5990:
5989:
5984:
5972:
5970:
5969:
5964:
5952:
5950:
5949:
5944:
5929:
5927:
5926:
5921:
5919:
5918:
5906:
5905:
5889:
5887:
5886:
5881:
5857:
5855:
5854:
5849:
5847:
5846:
5822:
5821:
5793:
5791:
5790:
5785:
5783:
5782:
5767:
5766:
5741:
5739:
5738:
5733:
5721:
5719:
5718:
5713:
5695:
5693:
5692:
5687:
5671:
5669:
5668:
5663:
5652:. One says that
5640:
5638:
5637:
5632:
5614:
5612:
5611:
5606:
5590:
5588:
5587:
5582:
5571:. One says that
5562:
5560:
5559:
5554:
5552:
5551:
5539:
5538:
5522:
5520:
5519:
5514:
5489:
5487:
5486:
5481:
5466:
5464:
5463:
5458:
5456:
5455:
5431:
5430:
5405:
5403:
5402:
5397:
5385:
5383:
5382:
5377:
5359:
5357:
5356:
5351:
5321:
5319:
5318:
5313:
5295:
5293:
5292:
5287:
5258:to a hypergraph
5253:
5251:
5250:
5245:
5191:Bachman diagrams
5167:guarded fragment
5132:
5130:
5129:
5124:
5122:
5111:
5090:
5088:
5087:
5082:
5074:
5053:
5051:
5050:
5045:
5043:
5022:
5020:
5019:
5014:
5012:
4961:
4959:
4958:
4953:
4951:
4948:
4941:
4894:
4893:
4881:
4880:
4866:
4865:
4855:
4854:
4853:
4852:
4827:
4826:
4805:
4803:
4802:
4797:
4795:
4787:
4786:
4785:
4784:
4763:
4761:
4760:
4755:
4753:
4752:
4730:
4728:
4727:
4722:
4717:
4716:
4683:adjacency matrix
4677:Adjacency matrix
4602:
4600:
4599:
4594:
4592:
4589:
4582:
4541:
4540:
4522:
4521:
4510:
4509:
4489:
4488:
4470:
4469:
4458:
4457:
4430:
4429:
4407:
4405:
4404:
4399:
4394:
4393:
4365:
4363:
4362:
4357:
4352:
4351:
4329:
4327:
4326:
4321:
4316:
4315:
4293:
4291:
4290:
4285:
4283:
4282:
4263:
4261:
4260:
4255:
4247:
4246:
4225:
4223:
4222:
4217:
4214:
4209:
4196:
4191:
4180:
4176:
4175:
4162:
4157:
4141:
4139:
4138:
4133:
4131:
4130:
4117:
4112:
4096:
4094:
4093:
4088:
4086:
4085:
4065:
4063:
4062:
4057:
4055:
4054:
4034:
4032:
4031:
4026:
4024:
4023:
4007:
4005:
4004:
3999:
3983:
3981:
3980:
3975:
3970:
3969:
3958:
3954:
3953:
3938:
3937:
3917:
3915:
3914:
3909:
3907:
3906:
3884:
3882:
3881:
3876:
3874:
3871:
3864:
3826:
3825:
3813:
3812:
3801:
3800:
3776:
3775:
3753:
3751:
3750:
3745:
3740:
3739:
3707:incidence matrix
3705:
3703:
3702:
3697:
3679:
3677:
3676:
3671:
3666:
3665:
3654:
3648:
3644:
3643:
3631:
3630:
3605:
3603:
3602:
3597:
3592:
3591:
3580:
3571:
3567:
3566:
3554:
3553:
3525:Incidence matrix
3432:
3430:
3429:
3424:
3413:
3412:
3407:
3403:
3402:
3371:
3369:
3368:
3363:
3355:
3354:
3342:
3341:
3329:
3328:
3313:
3312:
3293:
3291:
3290:
3285:
3280:
3279:
3260:
3258:
3257:
3252:
3247:
3246:
3227:
3225:
3224:
3219:
3207:
3205:
3204:
3199:
3197:
3196:
3174:
3172:
3171:
3166:
3161:
3157:
3147:
3146:
3134:
3133:
3115:
3114:
3065:
3063:
3062:
3057:
3036:
3034:
3033:
3028:
3023:
3019:
3003:
3002:
2969:
2967:
2966:
2961:
2949:
2947:
2946:
2941:
2939:
2938:
2908:
2906:
2905:
2900:
2892:
2851:
2836:
2834:
2833:
2828:
2816:
2798:
2783:
2781:
2780:
2775:
2770:
2759:
2736:
2735:
2707:
2705:
2704:
2699:
2694:
2693:
2668:
2666:
2665:
2660:
2658:
2657:
2641:
2639:
2638:
2633:
2600:
2598:
2597:
2592:
2587:
2583:
2523:
2522:
2503:
2501:
2500:
2495:
2477:
2475:
2474:
2469:
2467:
2466:
2439:
2437:
2436:
2431:
2429:
2428:
2412:
2410:
2409:
2404:
2402:
2401:
2382:
2380:
2379:
2374:
2360:
2359:
2341:
2340:
2328:
2327:
2309:
2308:
2274:
2272:
2271:
2266:
2258:
2257:
2239:
2238:
2210:
2208:
2207:
2202:
2098:
2096:
2095:
2090:
2077:
2075:
2074:
2069:
2049:
2047:
2046:
2041:
2025:
2023:
2022:
2017:
2003:
2001:
2000:
1995:
1980:
1978:
1977:
1972:
1958:
1956:
1955:
1950:
1898:
1896:
1895:
1890:
1878:
1876:
1875:
1870:
1856:
1854:
1853:
1848:
1781:
1779:
1778:
1773:
1622:Ramsey's theorem
1602:(also known as:
1571:money laundering
1520:machine learning
1498:with hypergraph
1473:cooperative game
1426:
1424:
1423:
1418:
1388:
1386:
1385:
1380:
1378:
1370:
1358:
1356:
1355:
1350:
1332:
1330:
1329:
1324:
1299:
1297:
1296:
1291:
1286:
1278:
1270:
1262:
1251:
1243:
1231:
1229:
1228:
1223:
1193:order of an edge
1190:
1188:
1187:
1182:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1098:
1096:
1095:
1090:
1071:is known as its
1070:
1068:
1067:
1062:
1040:
1038:
1037:
1032:
1002:
1000:
999:
994:
982:
980:
979:
974:
946:
944:
943:
938:
926:
924:
923:
918:
863:
861:
860:
855:
811:
809:
808:
803:
753:
751:
750:
745:
707:
705:
704:
699:
661:
659:
658:
653:
612:
610:
609:
604:
596:
595:
583:
582:
570:
569:
557:
556:
544:
543:
518:
516:
515:
510:
439:
437:
436:
431:
423:
422:
404:
402:
401:
396:
388:
387:
375:
374:
362:
361:
343:
341:
340:
335:
327:
326:
314:
313:
295:
293:
292:
287:
279:
278:
266:
265:
253:
252:
231:
229:
228:
223:
215:
214:
202:
201:
189:
188:
176:
175:
150:
148:
147:
142:
137:
136:
124:
123:
111:
110:
98:
97:
85:
84:
72:
71:
59:
58:
21:
10611:
10610:
10606:
10605:
10604:
10602:
10601:
10600:
10576:
10575:
10574:
10569:
10548:
10512:
10434:
10369:
10365:Incidence graph
10348:
10343:
10313:
10308:
10282:
10243:
10206:
10175:Data structures
10170:
10167:
10129:
10102:
10081:
10060:
10027:
10021:
10000:
9982:
9977:
9976:
9943:
9939:
9928:
9924:
9908:
9907:
9901:
9899:
9872:
9868:
9852:
9851:
9838:10.1.1.553.2367
9821:
9817:
9805:
9803:
9796:
9780:
9776:
9739:
9730:
9723:10.1137/0213035
9700:
9696:
9689:
9664:
9660:
9649:
9642:
9633:
9631:
9627:
9608:
9602:
9598:
9591:
9577:
9573:
9559:
9549:Matching Theory
9538:
9531:
9494:
9490:
9481:
9479:
9466:
9465:
9461:
9454:
9428:
9424:
9396:
9392:
9385:
9355:
9351:
9344:
9322:
9318:
9311:
9281:
9277:
9254:
9250:
9241:
9239:
9235:
9208:
9202:
9198:
9189:
9187:
9180:
9156:
9152:
9123:
9116:
9087:
9080:
9071:
9069:
9065:
9050:
9042:
9038:
9003:
8999:
8967:
8963:
8955:
8953:
8902:
8898:
8858:
8854:
8832:
8828:
8786:
8782:
8726:
8722:
8714:
8712:
8705:
8687:
8683:
8674:
8672:
8665:
8639:
8635:
8626:
8624:
8620:
8606:
8587:
8579:
8570:
8561:
8559:
8555:
8524:
8510:
8503:
8494:
8492:
8471:
8467:
8458:
8456:
8449:
8433:
8429:
8390:Haussler, David
8387:
8383:
8366:Ouvrard, Xavier
8363:
8359:
8350:
8348:
8344:
8300:(1). IEEE: 12.
8289:
8283:
8276:
8271:
8266:
8261:
8222:
8202:
8176:
8175:
8170:
8164:
8163:
8158:
8151:
8147:
8145:
8142:
8141:
8099:
8095:
8083:
8079:
8077:
8074:
8073:
8053:
8049:
8037:
8033:
8031:
8028:
8027:
8010:
8006:
8004:
8001:
8000:
7983:
7979:
7977:
7974:
7973:
7940:
7936:
7928:
7925:
7924:
7907:
7903:
7894:
7890:
7888:
7885:
7884:
7867:
7863:
7855:
7852:
7851:
7831:
7827:
7809:
7805:
7803:
7800:
7799:
7764:
7760:
7758:
7755:
7754:
7720:
7717:
7716:
7673:
7662:
7656:
7653:
7638:
7622:
7611:
7584:is the rank of
7560:
7557:
7556:
7515:
7511:
7510:
7506:
7502:
7493:
7482:
7476:
7473:
7472:
7441:
7438:
7437:
7420:
7416:
7414:
7411:
7410:
7391:
7387:
7386:
7382:
7380:
7377:
7376:
7360:
7357:
7356:
7333:
7329:
7314:
7310:
7301:
7297:
7292:
7289:
7288:
7248:
7245:
7244:
7241:
7226:edge-transitive
7197:
7193:
7181:
7177:
7169:
7166:
7165:
7144:
7140:
7138:
7135:
7134:
7117:
7113:
7111:
7108:
7107:
7076:
7073:
7072:
6995:
6992:
6991:
6966:
6963:
6962:
6957:
6956:
6952:
6930:
6926:
6917:
6913:
6911:
6908:
6907:
6885:
6882:
6881:
6865:
6862:
6861:
6845:
6842:
6841:
6815:
6811:
6802:
6798:
6789:
6785:
6783:
6780:
6779:
6760:
6757:
6756:
6724:
6720:
6711:
6707:
6698:
6694:
6692:
6689:
6688:
6669:
6666:
6665:
6649:
6646:
6645:
6610:
6607:
6606:
6590:
6587:
6586:
6570:
6567:
6566:
6525:
6521:
6494:
6490:
6463:
6459:
6432:
6428:
6401:
6397:
6370:
6366:
6355:
6352:
6351:
6307:
6303:
6276:
6272:
6245:
6241:
6214:
6210:
6183:
6179:
6152:
6148:
6137:
6134:
6133:
6114:
6111:
6110:
6107:
6048:
6038:
6034:
6030:
6029:
6027:
6024:
6023:
5998:
5995:
5994:
5978:
5975:
5974:
5958:
5955:
5954:
5938:
5935:
5934:
5914:
5910:
5901:
5897:
5895:
5892:
5891:
5890:if and only if
5869:
5866:
5865:
5833:
5829:
5817:
5813:
5805:
5802:
5801:
5778:
5774:
5762:
5758:
5750:
5747:
5746:
5727:
5724:
5723:
5701:
5698:
5697:
5681:
5678:
5677:
5657:
5654:
5653:
5620:
5617:
5616:
5600:
5597:
5596:
5576:
5573:
5572:
5547:
5543:
5534:
5530:
5528:
5525:
5524:
5523:if and only if
5502:
5499:
5498:
5475:
5472:
5471:
5442:
5438:
5426:
5422:
5414:
5411:
5410:
5391:
5388:
5387:
5371:
5368:
5367:
5333:
5330:
5329:
5301:
5298:
5297:
5263:
5260:
5259:
5221:
5218:
5217:
5207:
5195:polynomial time
5176:We can test in
5163:database schema
5159:database theory
5115:
5104:
5096:
5093:
5092:
5067:
5059:
5056:
5055:
5036:
5028:
5025:
5024:
5005:
4997:
4994:
4993:
4990:bipartite graph
4986:incidence graph
4967:
4946:
4945:
4913:
4911:
4905:
4904:
4889:
4885:
4876:
4872:
4858:
4856:
4848:
4844:
4843:
4839:
4835:
4831:
4819:
4815:
4813:
4810:
4809:
4791:
4780:
4776:
4775:
4771:
4769:
4766:
4765:
4742:
4738:
4736:
4733:
4732:
4709:
4705:
4694:
4691:
4690:
4679:
4671:incidence graph
4660:
4653:
4646:
4639:
4617:bipartite graph
4609:
4607:Incidence graph
4587:
4586:
4554:
4552:
4546:
4545:
4536:
4532:
4517:
4513:
4502:
4500:
4494:
4493:
4484:
4480:
4465:
4461:
4450:
4448:
4438:
4434:
4422:
4418:
4416:
4413:
4412:
4386:
4382:
4371:
4368:
4367:
4347:
4343:
4335:
4332:
4331:
4311:
4307:
4299:
4296:
4295:
4294:are denoted by
4278:
4274:
4272:
4269:
4268:
4239:
4235:
4233:
4230:
4229:
4210:
4205:
4192:
4187:
4171:
4167:
4158:
4153:
4147:
4144:
4143:
4126:
4122:
4113:
4108:
4102:
4099:
4098:
4081:
4077:
4075:
4072:
4071:
4050:
4046:
4044:
4041:
4040:
4019:
4015:
4013:
4010:
4009:
3993:
3990:
3989:
3965:
3961:
3949:
3945:
3933:
3929:
3927:
3924:
3923:
3902:
3898:
3896:
3893:
3892:
3869:
3868:
3836:
3834:
3828:
3827:
3821:
3817:
3808:
3804:
3793:
3791:
3784:
3780:
3768:
3764:
3762:
3759:
3758:
3732:
3728:
3717:
3714:
3713:
3685:
3682:
3681:
3661:
3657:
3639:
3635:
3626:
3622:
3611:
3608:
3607:
3587:
3583:
3562:
3558:
3549:
3545:
3534:
3531:
3530:
3527:
3438:connected graph
3408:
3398:
3394:
3390:
3389:
3387:
3384:
3383:
3350:
3346:
3337:
3333:
3324:
3320:
3308:
3304:
3302:
3299:
3298:
3275:
3271:
3266:
3263:
3262:
3242:
3238:
3233:
3230:
3229:
3213:
3210:
3209:
3192:
3188:
3186:
3183:
3182:
3142:
3138:
3129:
3125:
3110:
3106:
3096:
3092:
3078:
3075:
3074:
3045:
3042:
3041:
3040:Given a subset
2998:
2994:
2984:
2980:
2978:
2975:
2974:
2955:
2952:
2951:
2934:
2930:
2922:
2919:
2918:
2885:
2844:
2842:
2839:
2838:
2806:
2791:
2789:
2786:
2785:
2763:
2752:
2731:
2727:
2716:
2713:
2712:
2689:
2685:
2674:
2671:
2670:
2653:
2649:
2647:
2644:
2643:
2627:
2624:
2623:
2606:restriction of
2531:
2527:
2518:
2514:
2512:
2509:
2508:
2483:
2480:
2479:
2462:
2458:
2456:
2453:
2452:
2424:
2420:
2418:
2415:
2414:
2397:
2393:
2391:
2388:
2387:
2355:
2351:
2336:
2332:
2323:
2319:
2304:
2300:
2289:
2286:
2285:
2253:
2249:
2234:
2230:
2219:
2216:
2215:
2178:
2175:
2174:
2156:
2116:Downward-closed
2084:
2081:
2080:
2057:
2054:
2053:
2035:
2032:
2031:
2011:
2008:
2007:
1989:
1986:
1985:
1966:
1963:
1962:
1944:
1941:
1940:
1884:
1881:
1880:
1864:
1861:
1860:
1842:
1839:
1838:
1813:- has no edges.
1804:
1731:
1728:
1727:
1724:
1664:circuit diagram
1656:
1583:
1532:image retrieval
1512:
1454:bipartite graph
1437:drawn from the
1400:
1397:
1396:
1374:
1366:
1364:
1361:
1360:
1338:
1335:
1334:
1312:
1309:
1308:
1282:
1274:
1266:
1258:
1247:
1239:
1237:
1234:
1233:
1199:
1196:
1195:
1176:
1173:
1172:
1152:
1149:
1148:
1120:
1117:
1116:
1084:
1081:
1080:
1056:
1053:
1052:
1008:
1005:
1004:
988:
985:
984:
968:
965:
964:
932:
929:
928:
900:
897:
896:
816:
813:
812:
758:
755:
754:
712:
709:
708:
666:
663:
662:
617:
614:
613:
591:
587:
578:
574:
565:
561:
552:
548:
539:
535:
524:
521:
520:
462:
459:
458:
418:
414:
409:
406:
405:
383:
379:
370:
366:
357:
353:
348:
345:
344:
322:
318:
309:
305:
300:
297:
296:
274:
270:
261:
257:
248:
244:
236:
233:
232:
210:
206:
197:
193:
184:
180:
171:
167:
156:
153:
152:
132:
128:
119:
115:
106:
102:
93:
89:
80:
76:
67:
63:
54:
50:
39:
36:
35:
28:
23:
22:
15:
12:
11:
5:
10609:
10599:
10598:
10593:
10588:
10571:
10570:
10568:
10567:
10562:
10556:
10554:
10550:
10549:
10547:
10546:
10541:
10539:Beck's theorem
10536:
10531:
10526:
10520:
10518:
10514:
10513:
10511:
10510:
10505:
10500:
10495:
10490:
10485:
10480:
10475:
10470:
10465:
10460:
10455:
10450:
10444:
10442:
10440:Configurations
10436:
10435:
10433:
10432:
10431:
10430:
10422:
10421:
10420:
10412:
10411:
10410:
10405:
10395:
10394:
10393:
10391:Steiner system
10388:
10377:
10375:
10371:
10370:
10368:
10367:
10362:
10356:
10354:
10353:Representation
10350:
10349:
10342:
10341:
10334:
10327:
10319:
10310:
10309:
10307:
10306:
10301:
10296:
10294:Graph database
10290:
10288:
10284:
10283:
10281:
10280:
10275:
10269:
10263:
10257:
10251:
10249:
10245:
10244:
10242:
10241:
10236:
10231:
10226:
10223:
10220:
10214:
10212:
10208:
10207:
10205:
10204:
10199:
10194:
10189:
10187:Adjacency list
10184:
10178:
10176:
10172:
10171:
10166:
10165:
10158:
10151:
10143:
10137:
10136:
10128:
10127:External links
10125:
10124:
10123:
10106:
10100:
10085:
10079:
10064:
10058:
10043:
10025:
10019:
10004:
9998:
9981:
9978:
9975:
9974:
9960:10.1.1.67.2498
9953:(7): 673–693,
9937:
9922:
9866:
9815:
9794:
9774:
9753:(3): 514–550.
9728:
9717:(3): 566–579.
9707:Yannakakis, M.
9694:
9687:
9658:
9640:
9596:
9589:
9571:
9557:
9545:Plummer, M. D.
9541:Lovász, László
9529:
9508:(3): 514–550.
9488:
9459:
9452:
9422:
9411:(3): 309–325,
9390:
9383:
9349:
9342:
9316:
9309:
9275:
9264:(3): 177–185,
9248:
9221:(2): 141–157,
9196:
9178:
9150:
9114:
9078:
9036:
9017:(9): 2529–54.
8997:
8971:Bioinformatics
8961:
8916:(1): 363–376,
8896:
8862:Bioinformatics
8852:
8826:
8780:
8720:
8703:
8681:
8663:
8633:
8604:
8568:
8535:(3): 479–513.
8518:Yannakakis, M.
8501:
8465:
8447:
8427:
8406:(2): 127–151,
8381:
8357:
8273:
8272:
8270:
8267:
8265:
8264:
8255:
8249:
8243:
8237:
8231:
8225:
8216:
8210:
8203:
8201:
8198:
8197:
8196:
8185:
8181:
8174:
8171:
8169:
8166:
8165:
8162:
8159:
8157:
8154:
8153:
8150:
8128:directed graph
8107:
8102:
8098:
8094:
8091:
8086:
8082:
8061:
8056:
8052:
8048:
8045:
8040:
8036:
8013:
8009:
7986:
7982:
7943:
7939:
7935:
7932:
7910:
7906:
7902:
7897:
7893:
7870:
7866:
7862:
7859:
7839:
7834:
7830:
7826:
7823:
7820:
7817:
7812:
7808:
7787:
7784:
7781:
7778:
7775:
7772:
7767:
7763:
7742:
7739:
7736:
7733:
7730:
7727:
7724:
7675:
7674:
7625:
7623:
7616:
7610:
7607:
7573:
7570:
7567:
7564:
7553:
7552:
7541:
7538:
7535:
7532:
7529:
7525:
7518:
7514:
7509:
7505:
7501:
7496:
7491:
7488:
7485:
7481:
7457:
7454:
7451:
7448:
7445:
7423:
7419:
7394:
7390:
7385:
7364:
7353:
7352:
7341:
7336:
7332:
7328:
7325:
7322:
7317:
7313:
7309:
7304:
7300:
7296:
7270:
7267:
7264:
7261:
7258:
7255:
7252:
7240:
7237:
7200:
7196:
7192:
7189:
7184:
7180:
7176:
7173:
7147:
7143:
7120:
7116:
7095:
7092:
7089:
7086:
7083:
7080:
6999:
6979:
6976:
6973:
6970:
6951:
6948:
6933:
6929:
6925:
6920:
6916:
6895:
6892:
6889:
6869:
6849:
6838:
6837:
6826:
6823:
6818:
6814:
6810:
6805:
6801:
6797:
6792:
6788:
6764:
6753:
6752:
6741:
6738:
6735:
6732:
6727:
6723:
6719:
6714:
6710:
6706:
6701:
6697:
6673:
6653:
6629:
6626:
6623:
6620:
6617:
6614:
6594:
6574:
6563:
6562:
6551:
6548:
6545:
6542:
6539:
6536:
6533:
6528:
6524:
6520:
6517:
6514:
6511:
6508:
6505:
6502:
6497:
6493:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6466:
6462:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6435:
6431:
6427:
6424:
6421:
6418:
6415:
6412:
6409:
6404:
6400:
6396:
6393:
6390:
6387:
6384:
6381:
6378:
6373:
6369:
6365:
6362:
6359:
6345:
6344:
6333:
6330:
6327:
6324:
6321:
6318:
6315:
6310:
6306:
6302:
6299:
6296:
6293:
6290:
6287:
6284:
6279:
6275:
6271:
6268:
6265:
6262:
6259:
6256:
6253:
6248:
6244:
6240:
6237:
6234:
6231:
6228:
6225:
6222:
6217:
6213:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6186:
6182:
6178:
6175:
6172:
6169:
6166:
6163:
6160:
6155:
6151:
6147:
6144:
6141:
6118:
6106:
6103:
6071:
6070:
6059:
6056:
6051:
6046:
6041:
6037:
6033:
6008:
6005:
6002:
5982:
5962:
5942:
5931:
5930:
5917:
5913:
5909:
5904:
5900:
5879:
5876:
5873:
5859:
5858:
5845:
5842:
5839:
5836:
5832:
5828:
5825:
5820:
5816:
5812:
5809:
5795:
5794:
5781:
5777:
5773:
5770:
5765:
5761:
5757:
5754:
5731:
5711:
5708:
5705:
5685:
5661:
5648:, and also of
5630:
5627:
5624:
5604:
5580:
5565:
5564:
5550:
5546:
5542:
5537:
5533:
5512:
5509:
5506:
5479:
5470:The bijection
5468:
5467:
5454:
5451:
5448:
5445:
5441:
5437:
5434:
5429:
5425:
5421:
5418:
5395:
5375:
5361:
5360:
5349:
5346:
5343:
5340:
5337:
5311:
5308:
5305:
5296:, written as
5285:
5282:
5279:
5276:
5273:
5270:
5267:
5243:
5240:
5237:
5234:
5231:
5228:
5225:
5206:
5203:
5121:
5118:
5114:
5110:
5107:
5103:
5100:
5080:
5077:
5073:
5070:
5066:
5063:
5042:
5039:
5035:
5032:
5011:
5008:
5004:
5001:
4975:acyclic graphs
4966:
4963:
4950:
4944:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4912:
4910:
4907:
4906:
4903:
4900:
4897:
4892:
4888:
4884:
4879:
4875:
4871:
4864:
4861:
4857:
4851:
4847:
4842:
4838:
4837:
4834:
4830:
4825:
4822:
4818:
4794:
4790:
4783:
4779:
4774:
4751:
4748:
4745:
4741:
4720:
4715:
4712:
4708:
4704:
4701:
4698:
4678:
4675:
4658:
4651:
4644:
4637:
4608:
4605:
4604:
4603:
4591:
4585:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4553:
4551:
4548:
4547:
4544:
4539:
4535:
4531:
4528:
4525:
4520:
4516:
4508:
4505:
4501:
4499:
4496:
4495:
4492:
4487:
4483:
4479:
4476:
4473:
4468:
4464:
4456:
4453:
4449:
4447:
4444:
4441:
4440:
4437:
4433:
4428:
4425:
4421:
4397:
4392:
4389:
4385:
4381:
4378:
4375:
4366:respectively.
4355:
4350:
4346:
4342:
4339:
4319:
4314:
4310:
4306:
4303:
4281:
4277:
4253:
4250:
4245:
4242:
4238:
4227:if and only if
4213:
4208:
4204:
4200:
4195:
4190:
4186:
4179:
4174:
4170:
4166:
4161:
4156:
4152:
4129:
4125:
4121:
4116:
4111:
4107:
4084:
4080:
4053:
4049:
4022:
4018:
3997:
3973:
3968:
3964:
3957:
3952:
3948:
3944:
3941:
3936:
3932:
3905:
3901:
3886:
3885:
3873:
3867:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3835:
3833:
3830:
3829:
3824:
3820:
3816:
3811:
3807:
3799:
3796:
3792:
3790:
3787:
3786:
3783:
3779:
3774:
3771:
3767:
3743:
3738:
3735:
3731:
3727:
3724:
3721:
3695:
3692:
3689:
3669:
3664:
3660:
3653:
3647:
3642:
3638:
3634:
3629:
3625:
3621:
3618:
3615:
3595:
3590:
3586:
3579:
3576:
3570:
3565:
3561:
3557:
3552:
3548:
3544:
3541:
3538:
3526:
3523:
3434:
3433:
3422:
3419:
3416:
3411:
3406:
3401:
3397:
3393:
3373:
3372:
3361:
3358:
3353:
3349:
3345:
3340:
3336:
3332:
3327:
3323:
3319:
3316:
3311:
3307:
3283:
3278:
3274:
3270:
3250:
3245:
3241:
3237:
3217:
3195:
3191:
3176:
3175:
3164:
3160:
3156:
3153:
3150:
3145:
3141:
3137:
3132:
3128:
3124:
3121:
3118:
3113:
3109:
3105:
3102:
3099:
3095:
3091:
3088:
3085:
3082:
3055:
3052:
3049:
3038:
3037:
3026:
3022:
3018:
3015:
3012:
3009:
3006:
3001:
2997:
2993:
2990:
2987:
2983:
2959:
2937:
2933:
2929:
2926:
2911:
2910:
2898:
2895:
2891:
2888:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2850:
2847:
2826:
2823:
2820:
2815:
2812:
2809:
2805:
2801:
2797:
2794:
2773:
2769:
2766:
2762:
2758:
2755:
2751:
2748:
2745:
2742:
2739:
2734:
2730:
2726:
2723:
2720:
2697:
2692:
2688:
2684:
2681:
2678:
2656:
2652:
2631:
2602:
2601:
2590:
2586:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2530:
2526:
2521:
2517:
2504:is defined as
2493:
2490:
2487:
2465:
2461:
2427:
2423:
2400:
2396:
2384:
2383:
2372:
2369:
2366:
2363:
2358:
2354:
2350:
2347:
2344:
2339:
2335:
2331:
2326:
2322:
2318:
2315:
2312:
2307:
2303:
2299:
2296:
2293:
2276:
2275:
2264:
2261:
2256:
2252:
2248:
2245:
2242:
2237:
2233:
2229:
2226:
2223:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2160:subhypergraphs
2155:
2152:
2151:
2150:
2140:
2139:
2138:
2113:
2103:
2102:
2101:
2088:
2067:
2064:
2061:
2039:
2015:
1993:
1982:
1970:
1948:
1937:
1936:
1935:
1900:
1888:
1868:
1846:
1835:
1829:
1814:
1803:
1800:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1723:
1720:
1655:
1652:
1644:
1643:
1637:
1631:
1625:
1619:
1613:
1607:
1597:
1582:
1579:
1536:bioinformatics
1511:
1508:
1434:family of sets
1416:
1413:
1410:
1407:
1404:
1377:
1373:
1369:
1348:
1345:
1342:
1322:
1319:
1316:
1305:directed graph
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1254:
1250:
1246:
1242:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1180:
1156:
1136:
1133:
1130:
1127:
1124:
1088:
1060:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
992:
972:
936:
916:
913:
910:
907:
904:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
602:
599:
594:
590:
586:
581:
577:
573:
568:
564:
560:
555:
551:
547:
542:
538:
534:
531:
528:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
429:
426:
421:
417:
413:
394:
391:
386:
382:
378:
373:
369:
365:
360:
356:
352:
333:
330:
325:
321:
317:
312:
308:
304:
285:
282:
277:
273:
269:
264:
260:
256:
251:
247:
243:
240:
221:
218:
213:
209:
205:
200:
196:
192:
187:
183:
179:
174:
170:
166:
163:
160:
140:
135:
131:
127:
122:
118:
114:
109:
105:
101:
96:
92:
88:
83:
79:
75:
70:
66:
62:
57:
53:
49:
46:
43:
26:
9:
6:
4:
3:
2:
10608:
10597:
10594:
10592:
10589:
10587:
10584:
10583:
10581:
10566:
10563:
10561:
10558:
10557:
10555:
10551:
10545:
10542:
10540:
10537:
10535:
10532:
10530:
10527:
10525:
10522:
10521:
10519:
10515:
10509:
10506:
10504:
10501:
10499:
10496:
10494:
10491:
10489:
10486:
10484:
10481:
10479:
10476:
10474:
10471:
10469:
10466:
10464:
10461:
10459:
10456:
10454:
10451:
10449:
10446:
10445:
10443:
10441:
10437:
10429:
10426:
10425:
10423:
10419:
10416:
10415:
10414:Graph theory
10413:
10409:
10406:
10404:
10401:
10400:
10399:
10396:
10392:
10389:
10387:
10384:
10383:
10382:
10381:Combinatorics
10379:
10378:
10376:
10372:
10366:
10363:
10361:
10358:
10357:
10355:
10351:
10347:
10340:
10335:
10333:
10328:
10326:
10321:
10320:
10317:
10305:
10302:
10300:
10299:Graph drawing
10297:
10295:
10292:
10291:
10289:
10285:
10279:
10276:
10273:
10272:Newick format
10270:
10267:
10264:
10261:
10258:
10256:
10253:
10252:
10250:
10246:
10240:
10237:
10235:
10232:
10230:
10227:
10224:
10221:
10219:
10216:
10215:
10213:
10209:
10203:
10200:
10198:
10195:
10193:
10190:
10188:
10185:
10183:
10180:
10179:
10177:
10173:
10164:
10159:
10157:
10152:
10150:
10145:
10144:
10141:
10134:
10131:
10130:
10122:
10120:
10116:
10112:
10107:
10103:
10097:
10093:
10092:
10086:
10082:
10076:
10072:
10071:
10065:
10061:
10055:
10051:
10050:
10044:
10040:
10036:
10035:
10030:
10026:
10022:
10016:
10012:
10011:
10005:
10001:
9995:
9991:
9990:
9984:
9983:
9970:
9966:
9961:
9956:
9952:
9948:
9941:
9933:
9926:
9918:
9912:
9897:
9893:
9889:
9885:
9881:
9877:
9870:
9862:
9856:
9848:
9844:
9839:
9834:
9830:
9826:
9819:
9812:
9801:
9797:
9791:
9787:
9786:
9778:
9770:
9766:
9761:
9756:
9752:
9748:
9744:
9737:
9735:
9733:
9724:
9720:
9716:
9712:
9708:
9704:
9703:Tarjan, R. E.
9698:
9690:
9688:0-201-53771-0
9684:
9680:
9676:
9672:
9668:
9667:Abiteboul, S.
9662:
9654:
9647:
9645:
9626:
9622:
9618:
9614:
9607:
9600:
9592:
9590:0-7204-2450-X
9586:
9582:
9575:
9568:
9564:
9560:
9558:0-444-87916-1
9554:
9550:
9546:
9542:
9536:
9534:
9525:
9521:
9516:
9511:
9507:
9503:
9499:
9492:
9477:
9473:
9469:
9463:
9455:
9449:
9444:
9439:
9435:
9434:
9433:Graph Drawing
9426:
9418:
9414:
9410:
9406:
9405:
9400:
9394:
9386:
9380:
9375:
9370:
9366:
9365:
9364:Graph Drawing
9360:
9353:
9345:
9339:
9335:
9331:
9327:
9320:
9312:
9306:
9301:
9296:
9292:
9291:
9290:Graph Drawing
9286:
9279:
9271:
9267:
9263:
9259:
9252:
9234:
9229:
9224:
9220:
9216:
9215:
9207:
9200:
9185:
9181:
9175:
9171:
9167:
9166:
9161:
9154:
9145:
9140:
9136:
9132:
9128:
9121:
9119:
9109:
9104:
9100:
9096:
9092:
9085:
9083:
9064:
9060:
9056:
9049:
9048:
9040:
9032:
9028:
9024:
9020:
9016:
9012:
9008:
9001:
8994:
8990:
8985:
8980:
8976:
8972:
8965:
8951:
8947:
8943:
8939:
8935:
8931:
8927:
8923:
8919:
8915:
8911:
8907:
8900:
8893:
8889:
8884:
8879:
8875:
8871:
8867:
8863:
8856:
8849:
8845:
8841:
8837:
8830:
8823:
8819:
8815:
8811:
8807:
8803:
8799:
8795:
8791:
8784:
8777:
8773:
8769:
8765:
8761:
8757:
8753:
8749:
8744:
8739:
8736:(6): 066118,
8735:
8731:
8724:
8710:
8706:
8704:9780262256919
8700:
8696:
8692:
8685:
8670:
8666:
8660:
8656:
8652:
8648:
8644:
8637:
8619:
8615:
8611:
8607:
8601:
8597:
8593:
8586:
8585:
8577:
8575:
8573:
8554:
8550:
8546:
8542:
8538:
8534:
8530:
8523:
8519:
8516:; Maier, D.;
8515:
8508:
8506:
8490:
8486:
8482:
8478:
8477:
8469:
8454:
8450:
8444:
8440:
8439:
8431:
8423:
8419:
8414:
8409:
8405:
8401:
8400:
8395:
8391:
8385:
8376:
8371:
8367:
8361:
8343:
8339:
8335:
8331:
8327:
8323:
8319:
8315:
8311:
8307:
8303:
8299:
8295:
8288:
8281:
8279:
8274:
8259:
8256:
8253:
8250:
8247:
8244:
8241:
8238:
8235:
8232:
8229:
8226:
8220:
8217:
8214:
8211:
8208:
8205:
8204:
8183:
8179:
8172:
8167:
8160:
8155:
8148:
8140:
8139:
8138:
8136:
8131:
8129:
8125:
8121:
8100:
8096:
8089:
8084:
8080:
8054:
8050:
8043:
8038:
8034:
8011:
8007:
7984:
7980:
7969:
7967:
7963:
7962:partial order
7959:
7941:
7937:
7933:
7930:
7908:
7904:
7900:
7895:
7891:
7868:
7864:
7860:
7857:
7832:
7828:
7824:
7821:
7815:
7810:
7806:
7782:
7779:
7776:
7770:
7765:
7761:
7737:
7734:
7731:
7725:
7722:
7714:
7710:
7706:
7705:partial order
7701:
7699:
7695:
7691:
7687:
7683:
7671:
7668:
7660:
7650:
7646:
7642:
7636:
7635:
7631:
7626:This section
7624:
7620:
7615:
7614:
7606:
7604:
7600:
7596:
7592:
7589:
7587:
7568:
7562:
7536:
7530:
7527:
7523:
7516:
7512:
7507:
7503:
7499:
7494:
7489:
7486:
7483:
7479:
7471:
7470:
7469:
7455:
7452:
7449:
7446:
7443:
7421:
7417:
7409:generated by
7392:
7388:
7383:
7362:
7334:
7330:
7326:
7323:
7320:
7315:
7311:
7307:
7302:
7298:
7287:
7286:
7285:
7284:
7265:
7262:
7259:
7253:
7250:
7236:
7233:
7231:
7227:
7223:
7219:
7214:
7198:
7194:
7190:
7182:
7178:
7171:
7163:
7145:
7141:
7118:
7114:
7106:. Two edges
7093:
7090:
7084:
7078:
7070:
7066:
7062:
7058:
7055:Two vertices
7053:
7050:
7048:
7044:
7040:
7036:
7032:
7027:
7025:
7021:
7017:
7013:
6997:
6974:
6968:
6961:
6947:
6931:
6927:
6923:
6918:
6914:
6893:
6890:
6887:
6867:
6847:
6821:
6816:
6812:
6808:
6803:
6799:
6795:
6790:
6786:
6778:
6777:
6776:
6762:
6736:
6730:
6725:
6721:
6717:
6712:
6708:
6704:
6699:
6695:
6687:
6686:
6685:
6671:
6651:
6643:
6627:
6624:
6618:
6612:
6592:
6572:
6565:Then clearly
6543:
6540:
6537:
6531:
6526:
6522:
6518:
6512:
6509:
6506:
6500:
6495:
6491:
6487:
6481:
6478:
6475:
6469:
6464:
6460:
6456:
6450:
6447:
6444:
6438:
6433:
6429:
6425:
6419:
6416:
6413:
6407:
6402:
6398:
6394:
6388:
6385:
6382:
6376:
6371:
6367:
6360:
6357:
6350:
6349:
6348:
6325:
6322:
6319:
6313:
6308:
6304:
6300:
6294:
6291:
6288:
6282:
6277:
6273:
6269:
6263:
6260:
6257:
6251:
6246:
6242:
6238:
6232:
6229:
6226:
6220:
6215:
6211:
6207:
6201:
6198:
6195:
6189:
6184:
6180:
6176:
6170:
6167:
6164:
6158:
6153:
6149:
6142:
6139:
6132:
6131:
6130:
6116:
6102:
6100:
6096:
6092:
6088:
6084:
6080:
6076:
6073:A hypergraph
6057:
6054:
6049:
6044:
6039:
6035:
6031:
6022:
6021:
6020:
6006:
6003:
6000:
5993:, and writes
5980:
5960:
5940:
5915:
5911:
5907:
5902:
5898:
5877:
5874:
5871:
5864:
5863:
5862:
5840:
5834:
5830:
5826:
5818:
5814:
5807:
5800:
5799:
5798:
5779:
5775:
5771:
5763:
5759:
5752:
5745:
5744:
5743:
5729:
5709:
5706:
5703:
5696:, and writes
5683:
5675:
5659:
5651:
5647:
5642:
5628:
5625:
5622:
5602:
5594:
5578:
5570:
5548:
5544:
5540:
5535:
5531:
5510:
5507:
5504:
5497:
5496:
5495:
5493:
5477:
5449:
5443:
5439:
5435:
5427:
5423:
5416:
5409:
5408:
5407:
5393:
5373:
5366:
5347:
5341:
5338:
5335:
5328:
5327:
5326:
5325:
5309:
5306:
5303:
5280:
5277:
5274:
5268:
5265:
5257:
5238:
5235:
5232:
5226:
5223:
5216:A hypergraph
5214:
5212:
5209:A hypergraph
5202:
5198:
5196:
5192:
5187:
5181:
5179:
5174:
5172:
5168:
5164:
5160:
5156:
5152:
5148:
5147:GYO algorithm
5144:
5138:
5136:
5119:
5116:
5112:
5108:
5105:
5101:
5098:
5078:
5075:
5071:
5068:
5064:
5061:
5040:
5037:
5033:
5030:
5009:
5006:
5002:
4999:
4991:
4987:
4983:
4978:
4976:
4972:
4962:
4942:
4908:
4901:
4898:
4890:
4886:
4882:
4877:
4873:
4849:
4845:
4840:
4832:
4828:
4823:
4820:
4816:
4807:
4788:
4781:
4777:
4772:
4749:
4746:
4743:
4739:
4713:
4710:
4706:
4699:
4696:
4688:
4684:
4674:
4672:
4667:
4665:
4661:
4654:
4647:
4640:
4633:
4629:
4625:
4621:
4618:
4614:
4611:A hypergraph
4583:
4549:
4537:
4533:
4526:
4523:
4518:
4514:
4497:
4485:
4481:
4474:
4471:
4466:
4462:
4445:
4442:
4435:
4431:
4426:
4423:
4419:
4411:
4410:
4409:
4390:
4387:
4383:
4376:
4373:
4348:
4344:
4337:
4312:
4308:
4301:
4279:
4275:
4265:
4251:
4248:
4243:
4240:
4236:
4228:
4211:
4206:
4202:
4198:
4193:
4188:
4184:
4177:
4172:
4168:
4164:
4159:
4154:
4150:
4127:
4123:
4119:
4114:
4109:
4105:
4082:
4078:
4069:
4051:
4047:
4038:
4020:
4016:
3995:
3987:
3966:
3962:
3955:
3950:
3946:
3939:
3934:
3930:
3921:
3903:
3899:
3891:
3865:
3831:
3822:
3818:
3814:
3809:
3805:
3788:
3781:
3777:
3772:
3769:
3765:
3757:
3756:
3755:
3736:
3733:
3729:
3722:
3719:
3710:
3708:
3693:
3690:
3687:
3662:
3658:
3651:
3645:
3640:
3636:
3632:
3627:
3623:
3616:
3613:
3588:
3584:
3577:
3574:
3568:
3563:
3559:
3555:
3550:
3546:
3539:
3536:
3522:
3520:
3519:Gaifman graph
3516:
3512:
3508:
3504:
3499:
3497:
3493:
3489:
3485:
3481:
3477:
3473:
3469:
3465:
3461:
3458:
3454:
3450:
3446:
3442:
3439:
3420:
3417:
3414:
3409:
3404:
3399:
3395:
3391:
3382:
3381:
3380:
3378:
3359:
3351:
3347:
3343:
3338:
3334:
3330:
3325:
3321:
3314:
3309:
3305:
3297:
3296:
3295:
3276:
3272:
3243:
3239:
3215:
3193:
3189:
3181:
3162:
3158:
3151:
3148:
3143:
3139:
3135:
3130:
3126:
3122:
3119:
3116:
3111:
3107:
3100:
3097:
3093:
3089:
3086:
3083:
3080:
3073:
3072:
3071:
3069:
3053:
3050:
3047:
3024:
3020:
3013:
3010:
3007:
3004:
2999:
2995:
2988:
2985:
2981:
2973:
2972:
2971:
2957:
2935:
2931:
2927:
2924:
2916:
2889:
2886:
2882:
2879:
2873:
2870:
2867:
2864:
2861:
2858:
2852:
2848:
2845:
2824:
2818:
2813:
2810:
2807:
2803:
2799:
2795:
2792:
2767:
2764:
2760:
2756:
2753:
2749:
2746:
2740:
2732:
2728:
2721:
2718:
2711:
2710:
2709:
2690:
2686:
2679:
2676:
2654:
2650:
2629:
2621:
2616:
2614:
2613:
2609:
2588:
2584:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2535:
2532:
2528:
2524:
2519:
2515:
2507:
2506:
2505:
2491:
2488:
2485:
2463:
2459:
2450:
2449:subhypergraph
2445:
2443:
2425:
2421:
2398:
2394:
2370:
2361:
2356:
2352:
2348:
2345:
2342:
2337:
2333:
2329:
2324:
2320:
2316:
2313:
2310:
2305:
2301:
2294:
2291:
2284:
2283:
2282:
2281:
2262:
2254:
2250:
2246:
2243:
2240:
2235:
2231:
2224:
2221:
2214:
2213:
2212:
2195:
2192:
2189:
2183:
2180:
2171:
2169:
2165:
2161:
2148:
2144:
2141:
2136:
2135:
2130:
2126:
2125:
2123:
2122:
2117:
2114:
2111:
2107:
2104:
2099:
2086:
2065:
2062:
2059:
2051:
2037:
2028:
2027:
2013:
2005:
1991:
1983:
1968:
1960:
1946:
1938:
1933:
1932:
1927:
1926:
1921:
1920:
1918:
1917:
1912:
1908:
1904:
1901:
1886:
1866:
1858:
1844:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1808:
1807:
1799:
1795:
1793:
1789:
1784:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1719:
1715:
1712:
1708:
1707:planar graphs
1704:
1700:
1697:
1687:
1683:
1681:
1676:
1675:graph drawing
1671:
1665:
1660:
1651:
1649:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1601:
1598:
1595:
1592:
1591:
1590:
1588:
1578:
1576:
1572:
1567:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1517:
1507:
1505:
1501:
1500:homomorphisms
1497:
1492:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1457:
1455:
1451:
1447:
1442:
1440:
1439:universal set
1436:
1435:
1430:
1411:
1408:
1405:
1395:
1390:
1371:
1346:
1343:
1340:
1320:
1317:
1314:
1306:
1301:
1279:
1271:
1263:
1252:
1244:
1216:
1213:
1210:
1204:
1201:
1194:
1178:
1170:
1154:
1131:
1128:
1125:
1115:
1110:
1108:
1107:
1102:
1086:
1078:
1074:
1058:
1050:
1049:
1044:
1041:is called an
1028:
1025:
1019:
1016:
1013:
990:
970:
962:
958:
954:
950:
934:
911:
908:
905:
894:
889:
887:
883:
879:
875:
871:
842:
836:
830:
827:
824:
799:
790:
787:
784:
778:
772:
769:
766:
741:
732:
726:
720:
695:
686:
680:
674:
649:
640:
634:
628:
600:
592:
588:
584:
579:
575:
571:
566:
562:
558:
553:
549:
545:
540:
536:
529:
526:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
467:
464:
455:
446:
419:
415:
392:
384:
380:
376:
371:
367:
363:
358:
354:
331:
323:
319:
315:
310:
306:
283:
275:
271:
267:
262:
258:
254:
249:
245:
219:
211:
207:
203:
198:
194:
190:
185:
181:
177:
172:
168:
161:
158:
133:
129:
125:
120:
116:
112:
107:
103:
99:
94:
90:
86:
81:
77:
73:
68:
64:
60:
55:
51:
44:
41:
32:
19:
18:Gaifman graph
10553:Applications
10417:
10386:Block design
10266:LCF notation
10108:
10090:
10069:
10052:. Springer.
10048:
10032:
10029:"Hypergraph"
10009:
9992:. Elsevier.
9988:
9950:
9946:
9940:
9931:
9925:
9900:, retrieved
9883:
9879:
9869:
9831:(1): 69–79,
9828:
9824:
9818:
9810:
9804:. Retrieved
9785:Graph Theory
9784:
9777:
9750:
9746:
9714:
9710:
9697:
9678:
9661:
9652:
9632:. Retrieved
9612:
9599:
9580:
9574:
9548:
9505:
9501:
9491:
9480:. Retrieved
9471:
9462:
9432:
9425:
9408:
9402:
9393:
9363:
9352:
9325:
9319:
9289:
9285:Eades, Peter
9278:
9261:
9257:
9251:
9240:, retrieved
9218:
9212:
9199:
9188:, retrieved
9164:
9153:
9134:
9130:
9098:
9094:
9070:. Retrieved
9046:
9039:
9014:
9010:
9000:
8974:
8970:
8964:
8954:, retrieved
8913:
8909:
8899:
8865:
8861:
8855:
8839:
8835:
8829:
8797:
8793:
8783:
8733:
8729:
8723:
8713:, retrieved
8694:
8684:
8673:. Retrieved
8646:
8636:
8625:. Retrieved
8583:
8560:. Retrieved
8532:
8528:
8493:. Retrieved
8475:
8468:
8457:. Retrieved
8437:
8430:
8403:
8397:
8384:
8360:
8349:. Retrieved
8297:
8293:
8219:Factor graph
8132:
7970:
7702:
7694:term algebra
7682:ad infinitum
7681:
7678:
7663:
7657:January 2021
7654:
7639:Please help
7627:
7593:
7590:
7585:
7554:
7354:
7242:
7234:
7229:
7225:
7221:
7217:
7215:
7161:
7068:
7064:
7060:
7056:
7054:
7051:
7046:
7042:
7038:
7034:
7033:of a vertex
7030:
7028:
7024:k-hypergraph
7023:
7019:
7015:
7011:
6955:
6953:
6839:
6754:
6641:
6564:
6346:
6108:
6098:
6086:
6082:
6078:
6075:automorphism
6072:
5932:
5860:
5796:
5673:
5649:
5645:
5643:
5592:
5568:
5566:
5469:
5362:
5255:
5215:
5211:homomorphism
5208:
5199:
5186:Ronald Fagin
5182:
5175:
5139:
4982:Claude Berge
4979:
4968:
4808:
4680:
4668:
4663:
4656:
4649:
4642:
4635:
4631:
4627:
4623:
4619:
4612:
4610:
4266:
4067:
4036:
3985:
3887:
3711:
3528:
3518:
3515:primal graph
3514:
3510:
3507:clique graph
3506:
3502:
3500:
3495:
3491:
3487:
3483:
3479:
3471:
3467:
3463:
3456:
3452:
3448:
3444:
3440:
3435:
3374:
3179:
3177:
3067:
3039:
2914:
2912:
2619:
2617:
2611:
2607:
2605:
2603:
2448:
2446:
2385:
2279:
2277:
2172:
2167:
2163:
2159:
2157:
2142:
2132:
2131:is called a
2128:
2119:
2115:
2109:
2105:
2079:
2030:
1984:
1939:
1930:
1924:
1914:
1910:
1906:
1902:
1837:
1831:
1825:
1822:
1819:
1816:
1810:
1805:
1796:
1791:
1787:
1785:
1725:
1716:
1702:
1699:Venn diagram
1695:
1692:
1672:
1669:
1645:
1603:
1584:
1568:
1563:
1559:
1555:
1552:Apache Spark
1513:
1510:Applications
1493:
1488:
1484:
1477:simple games
1476:
1468:
1464:
1458:
1443:
1432:
1428:
1393:
1391:
1302:
1192:
1168:
1113:
1111:
1104:
1100:
1076:
1072:
1046:
1042:
960:
956:
952:
948:
892:
891:Formally, a
890:
880:in which an
873:
867:
10586:Hypergraphs
10424:Statistics
10304:Linked data
9671:Hull, R. B.
9615:: 306–312.
9101:: 293–306.
8512:Beeri, C.;
7067:are called
7029:The degree
6129:with edges
5646:equivalence
5492:isomorphism
5365:permutation
5178:linear time
5135:linear time
3984:called the
2708:. Formally
2478:induced by
2278:and having
2078:) is both
1903:2-colorable
1899:hyperedges.
1792:k-colorable
1711:NP-complete
1604:transversal
1465:range space
870:mathematics
10580:Categories
10453:Fano plane
10418:Hypergraph
10115:PlanetMath
10111:hypergraph
9980:References
9902:2018-10-13
9806:2021-06-12
9634:2018-09-02
9482:2022-04-27
9242:2010-05-17
9190:2010-05-17
9072:2021-01-20
8956:2017-09-22
8715:2021-07-24
8675:2021-01-20
8627:2021-01-08
8562:2021-01-03
8495:2021-01-20
8459:2021-06-12
8394:Welzl, Emo
8375:2002.05014
8351:2020-09-08
8240:Multigraph
8137:is simply
7239:Partitions
7230:transitive
5861:Note that
5674:equivalent
5406:such that
5256:isomorphic
3449:host graph
3377:involution
2442:index sets
1916:Property B
1817:Non-simple
1489:connectors
1485:hyperlinks
1450:Levi graph
1429:set system
895:is a pair
874:hypergraph
10403:Incidence
10274:for trees
10192:Edge list
10039:EMS Press
9955:CiteSeerX
9833:CiteSeerX
9675:Vianu, V.
9524:0004-5411
8743:0903.0419
8514:Fagin, R.
8338:199518871
8322:1077-2626
8314:1941-0506
8258:Petri Net
8124:bipartite
7934:∈
7901:∈
7861:∈
7628:does not
7480:∑
7453:≤
7447:≤
7324:⋯
7283:partition
7172:ϕ
7162:symmetric
7079:ϕ
7069:symmetric
7043:k-regular
7020:k-uniform
6932:∗
6924:≅
6919:∗
6891:≡
6825:∅
6809:∩
6796:∩
6755:In graph
6718:∩
6705:∩
6664:, vertex
6628:α
6613:ϕ
6544:δ
6538:β
6513:γ
6507:α
6482:α
6476:δ
6451:δ
6445:γ
6420:γ
6414:β
6389:β
6383:α
6050:∗
6040:∗
5941:π
5916:∗
5908:≅
5903:∗
5875:≡
5835:π
5808:ϕ
5753:ϕ
5730:ϕ
5707:≡
5626:≅
5549:∗
5541:≃
5536:∗
5508:≃
5478:ϕ
5444:π
5417:ϕ
5374:π
5345:→
5336:ϕ
5324:bijection
5307:≃
5151:confluent
5113:∈
5076:∈
5034:≠
5003:≠
4899:∈
4789:∈
4747:≤
4524:∈
4472:∈
4443:−
4212:∗
4199:∈
4194:∗
4173:∗
4165:∈
4160:∗
4128:∗
4120:∈
4115:∗
4083:∗
4052:∗
4021:∗
3967:∗
3951:∗
3935:∗
3920:incidence
3890:transpose
3815:∈
3691:×
3652:…
3575:…
3503:2-section
3410:∗
3400:∗
3344:∈
3331:∣
3194:∗
3149:⊆
3123:∈
3117:∣
3084:×
3051:⊆
3011:∈
3005:∣
2928:⊂
2883:∪
2874:⊆
2868:∣
2862:∈
2822:∖
2811:∈
2804:⋃
2750:∪
2578:∅
2575:≠
2569:∩
2557:∈
2551:∣
2545:∩
2489:⊆
2365:∅
2362:≠
2343:⊆
2317:∈
2311:∣
2247:∈
2241:∣
2110:reduction
2063:≥
1981:vertices.
1925:bipartite
1767:λ
1504:morphisms
1344:⊆
1318:⊆
1048:hyperedge
1026:∈
10517:Theorems
10428:Blocking
10398:Geometry
9911:citation
9896:archived
9855:citation
9800:Archived
9677:(1995).
9625:Archived
9547:(1986),
9476:Archived
9233:archived
9184:archived
9063:Archived
9031:11290643
8993:19648139
8950:archived
8938:22692911
8892:23812989
8768:19658575
8709:archived
8669:Archived
8618:Archived
8553:Archived
8520:(1983).
8489:Archived
8453:Archived
8342:Archived
8330:31398121
8246:P system
8228:Greedoid
8207:BF-graph
8200:See also
7709:preorder
6950:Symmetry
6105:Examples
6089:)) is a
5650:equality
5120:′
5109:′
5072:′
5041:′
5010:′
4687:adjacent
4008:, where
3379:, i.e.,
2890:′
2849:′
2796:′
2768:′
2757:′
2440:are the
2280:edge set
2050:-partite
2004:-partite
1959:-uniform
1931:balanced
1857:-regular
1823:multiple
1709:, it is
1587:theorems
1496:category
1106:codomain
1099:as its
961:elements
953:vertices
927:, where
886:vertices
10229:GraphML
10133:PAOHVis
10041:, 2001
9567:0859549
8946:7432373
8918:Bibcode
8883:3694678
8802:Bibcode
8776:6391099
8748:Bibcode
8614:5130573
8549:2418740
8422:0884223
7649:removed
7634:sources
7016:uniform
6085:,
5973:equals
5143:chordal
4634:, and (
3918:of the
3482:and of
3460:induces
2143:Laminar
2134:matroid
2106:Reduced
1079:, and
10374:Fields
10098:
10077:
10056:
10017:
9996:
9957:
9835:
9792:
9769:597990
9767:
9685:
9587:
9565:
9555:
9522:
9450:
9381:
9340:
9307:
9176:
9029:
8991:
8944:
8936:
8890:
8880:
8822:432036
8820:
8774:
8766:
8701:
8661:
8612:
8602:
8547:
8445:
8420:
8336:
8328:
8320:
8312:
7707:nor a
7555:where
5363:and a
4971:cycles
4965:Cycles
4867:
4806:with
4511:
4459:
4408:where
4181:
4097:. For
4066:is an
4035:is an
3959:
3802:
3754:where
3655:
3649:
3581:
3572:
3294:where
3066:, the
2386:where
2029:Every
1832:Simple
1469:ranges
1191:. The
1167:. The
1077:domain
957:points
449:edges.
10262:(GML)
10239:XGMML
10222:DotML
9765:S2CID
9628:(PDF)
9609:(PDF)
9236:(PDF)
9209:(PDF)
9066:(PDF)
9051:(PDF)
9027:S2CID
8942:S2CID
8818:S2CID
8772:S2CID
8738:arXiv
8621:(PDF)
8610:S2CID
8588:(PDF)
8556:(PDF)
8545:S2CID
8525:(PDF)
8370:arXiv
8345:(PDF)
8334:S2CID
8310:eISSN
8290:(PDF)
8269:Notes
7698:terms
6091:group
4988:(the
3447:is a
2784:with
1811:Empty
1662:This
1585:Many
1471:. In
1431:or a
959:, or
949:nodes
878:graph
10508:Dual
10225:GEXF
10218:DGML
10096:ISBN
10075:ISBN
10054:ISBN
10015:ISBN
9994:ISBN
9917:link
9861:link
9790:ISBN
9683:ISBN
9585:ISBN
9553:ISBN
9520:ISSN
9448:ISBN
9379:ISBN
9338:ISBN
9305:ISBN
9174:ISBN
8989:PMID
8934:PMID
8888:PMID
8764:PMID
8699:ISBN
8659:ISBN
8600:ISBN
8443:ISBN
8326:PMID
8318:ISSN
8072:and
7999:and
7883:and
7798:and
7632:any
7630:cite
7220:(or
7133:and
7059:and
7031:d(v)
6958:rank
6954:The
6860:and
6642:etc.
6585:and
6347:and
6081:(= (
5797:and
5742:has
5155:ears
5091:and
4973:and
4626:and
4330:and
4142:and
3986:dual
3888:The
3606:and
3529:Let
3505:(or
3501:The
3451:for
3180:dual
3178:The
2913:The
2837:and
2413:and
2173:Let
2166:and
1928:and
1909:and
1112:The
1101:head
1073:tail
1043:edge
963:and
882:edge
872:, a
519:and
151:and
10255:DOT
10234:GXL
10113:on
9965:doi
9888:doi
9843:doi
9755:doi
9719:doi
9617:doi
9510:doi
9438:doi
9413:doi
9369:doi
9330:doi
9295:doi
9266:doi
9223:doi
9139:doi
9103:doi
9099:658
9055:doi
9019:doi
8979:doi
8926:doi
8878:PMC
8870:doi
8844:doi
8810:doi
8756:doi
8651:doi
8592:doi
8537:doi
8481:doi
8408:doi
8302:doi
7643:by
7063:of
7041:is
7018:or
6101:).
5676:to
5672:is
5595:to
5591:is
5386:of
5254:is
5169:of
4662:in
4641:,
3988:of
3490:of
3478:of
3208:of
2618:An
2610:to
1820:(or
1502:as
1487:or
1392:An
1333:or
1103:or
1075:or
1045:or
868:In
10582::
10037:,
10031:,
9963:,
9951:10
9949:,
9913:}}
9909:{{
9894:,
9884:26
9878:,
9857:}}
9853:{{
9841:,
9827:,
9809:.
9798:.
9763:.
9751:30
9749:.
9745:.
9731:^
9715:13
9713:.
9705:;
9673:;
9669:;
9643:^
9623:.
9611:.
9563:MR
9561:,
9543:;
9532:^
9518:.
9506:30
9504:.
9500:.
9474:.
9470:.
9446:,
9409:11
9407:,
9377:,
9336:,
9303:,
9262:34
9260:,
9231:,
9219:10
9217:,
9211:,
9182:,
9168:,
9162:,
9135:42
9133:.
9129:.
9117:^
9097:.
9093:.
9081:^
9061:.
9025:.
9015:61
9013:.
9009:.
8987:,
8975:25
8973:,
8948:,
8940:,
8932:,
8924:,
8914:22
8912:,
8908:,
8886:,
8876:,
8866:29
8864:,
8840:44
8838:,
8816:,
8808:,
8798:7S
8796:,
8792:,
8770:,
8762:,
8754:,
8746:,
8734:79
8732:,
8707:,
8693:,
8667:.
8657:.
8645:.
8616:.
8608:.
8598:.
8571:^
8551:.
8543:.
8533:30
8531:.
8527:.
8504:^
8487:.
8451:.
8418:MR
8416:,
8402:,
8392:;
8340:.
8332:.
8324:.
8316:.
8308:.
8298:26
8296:.
8292:.
8277:^
8130:.
7968:.
7588:.
7232:.
7213:.
7049:.
6946:.
6640:,
5197:.
5173:.
4673:.
4666:.
4632:BG
4620:BG
4264:.
3709:.
3517:,
3513:,
3509:,
3498:.
3496:H'
3488:G'
3470:,
3436:A
2615:.
2447:A
2170:.
2162:,
1919:.
1606:);
1577:.
1506:.
1491:.
1441:.
1109:.
955:,
951:,
864:.
10338:e
10331:t
10324:v
10162:e
10155:t
10148:v
10121:.
10104:.
10083:.
10062:.
10023:.
10002:.
9971:.
9967::
9919:)
9905:.
9890::
9863:)
9849:.
9845::
9829:7
9771:.
9757::
9725:.
9721::
9691:.
9637:.
9619::
9593:.
9526:.
9512::
9485:.
9457:.
9440::
9420:.
9415::
9388:.
9371::
9347:.
9332::
9314:.
9297::
9273:.
9268::
9246:.
9225::
9194:.
9147:.
9141::
9111:.
9105::
9075:.
9057::
9033:.
9021::
8981::
8928::
8920::
8872::
8846::
8812::
8804::
8758::
8750::
8740::
8678:.
8653::
8630:.
8594::
8565:.
8539::
8498:.
8483::
8462:.
8425:.
8410::
8404:2
8379:.
8372::
8354:.
8304::
8184:.
8180:]
8173:0
8168:1
8161:1
8156:0
8149:[
8106:}
8101:1
8097:e
8093:{
8090:=
8085:2
8081:e
8060:}
8055:2
8051:e
8047:{
8044:=
8039:1
8035:e
8012:2
8008:e
7985:1
7981:e
7942:2
7938:e
7931:b
7909:2
7905:e
7896:1
7892:e
7869:1
7865:e
7858:b
7838:}
7833:1
7829:e
7825:,
7822:a
7819:{
7816:=
7811:2
7807:e
7786:}
7783:b
7780:,
7777:a
7774:{
7771:=
7766:1
7762:e
7741:}
7738:b
7735:,
7732:a
7729:{
7726:=
7723:V
7670:)
7664:(
7659:)
7655:(
7651:.
7637:.
7586:H
7572:)
7569:H
7566:(
7563:r
7540:)
7537:H
7534:(
7531:r
7528:=
7524:)
7517:k
7513:X
7508:H
7504:(
7500:r
7495:K
7490:1
7487:=
7484:k
7456:K
7450:k
7444:1
7422:k
7418:X
7393:k
7389:X
7384:H
7363:X
7340:)
7335:K
7331:X
7327:,
7321:,
7316:2
7312:X
7308:,
7303:1
7299:X
7295:(
7269:)
7266:E
7263:,
7260:X
7257:(
7254:=
7251:H
7199:j
7195:e
7191:=
7188:)
7183:i
7179:e
7175:(
7146:j
7142:e
7119:i
7115:e
7094:y
7091:=
7088:)
7085:x
7082:(
7065:H
7061:y
7057:x
7047:k
7039:H
7035:v
7012:k
6998:H
6978:)
6975:H
6972:(
6969:r
6928:G
6915:H
6894:G
6888:H
6868:G
6848:H
6822:=
6817:6
6813:f
6804:4
6800:f
6791:1
6787:f
6763:G
6740:}
6737:a
6734:{
6731:=
6726:6
6722:e
6713:4
6709:e
6700:1
6696:e
6672:a
6652:H
6625:=
6622:)
6619:a
6616:(
6593:G
6573:H
6550:}
6547:}
6541:,
6535:{
6532:=
6527:6
6523:f
6519:,
6516:}
6510:,
6504:{
6501:=
6496:5
6492:f
6488:,
6485:}
6479:,
6473:{
6470:=
6465:4
6461:f
6457:,
6454:}
6448:,
6442:{
6439:=
6434:3
6430:f
6426:,
6423:}
6417:,
6411:{
6408:=
6403:2
6399:f
6395:,
6392:}
6386:,
6380:{
6377:=
6372:1
6368:f
6364:{
6361:=
6358:G
6332:}
6329:}
6326:c
6323:,
6320:a
6317:{
6314:=
6309:6
6305:e
6301:,
6298:}
6295:d
6292:,
6289:b
6286:{
6283:=
6278:5
6274:e
6270:,
6267:}
6264:a
6261:,
6258:d
6255:{
6252:=
6247:4
6243:e
6239:,
6236:}
6233:d
6230:,
6227:c
6224:{
6221:=
6216:3
6212:e
6208:,
6205:}
6202:c
6199:,
6196:b
6193:{
6190:=
6185:2
6181:e
6177:,
6174:}
6171:b
6168:,
6165:a
6162:{
6159:=
6154:1
6150:e
6146:{
6143:=
6140:H
6117:H
6099:H
6087:E
6083:X
6079:H
6058:H
6055:=
6045:)
6036:H
6032:(
6007:G
6004:=
6001:H
5981:G
5961:H
5912:G
5899:H
5878:G
5872:H
5844:)
5841:i
5838:(
5831:f
5827:=
5824:)
5819:i
5815:e
5811:(
5780:n
5776:y
5772:=
5769:)
5764:n
5760:x
5756:(
5710:G
5704:H
5684:G
5660:H
5629:G
5623:H
5603:G
5579:H
5563:.
5545:G
5532:H
5511:G
5505:H
5453:)
5450:i
5447:(
5440:f
5436:=
5433:)
5428:i
5424:e
5420:(
5394:I
5348:Y
5342:X
5339::
5310:G
5304:H
5284:)
5281:F
5278:,
5275:Y
5272:(
5269:=
5266:G
5242:)
5239:E
5236:,
5233:X
5230:(
5227:=
5224:H
5117:f
5106:v
5102:,
5099:v
5079:f
5069:v
5065:,
5062:v
5038:f
5031:f
5007:v
5000:v
4943:.
4939:e
4936:s
4933:i
4930:w
4927:r
4924:e
4921:h
4918:t
4915:o
4909:0
4902:E
4896:)
4891:j
4887:v
4883:,
4878:i
4874:v
4870:(
4863:f
4860:i
4850:k
4846:e
4841:w
4833:{
4829:=
4824:j
4821:i
4817:a
4793:R
4782:k
4778:e
4773:w
4750:m
4744:k
4740:e
4719:)
4714:j
4711:i
4707:a
4703:(
4700:=
4697:A
4664:H
4659:1
4657:e
4652:1
4650:x
4645:1
4643:e
4638:1
4636:x
4628:E
4624:X
4613:H
4584:.
4580:e
4577:s
4574:i
4571:w
4568:r
4565:e
4562:h
4559:t
4556:o
4550:0
4543:)
4538:j
4534:e
4530:(
4527:H
4519:i
4515:v
4507:f
4504:i
4498:1
4491:)
4486:j
4482:e
4478:(
4475:T
4467:i
4463:v
4455:f
4452:i
4446:1
4436:{
4432:=
4427:j
4424:i
4420:b
4396:)
4391:j
4388:i
4384:b
4380:(
4377:=
4374:I
4354:)
4349:j
4345:e
4341:(
4338:T
4318:)
4313:j
4309:e
4305:(
4302:H
4280:j
4276:e
4252:1
4249:=
4244:j
4241:i
4237:b
4207:i
4203:e
4189:j
4185:v
4178:,
4169:E
4155:i
4151:e
4124:V
4110:j
4106:v
4079:V
4068:n
4048:E
4037:m
4017:V
3996:H
3972:)
3963:E
3956:,
3947:V
3943:(
3940:=
3931:H
3904:t
3900:I
3866:.
3862:e
3859:s
3856:i
3853:w
3850:r
3847:e
3844:h
3841:t
3838:o
3832:0
3823:j
3819:e
3810:i
3806:v
3798:f
3795:i
3789:1
3782:{
3778:=
3773:j
3770:i
3766:b
3742:)
3737:j
3734:i
3730:b
3726:(
3723:=
3720:I
3694:m
3688:n
3668:}
3663:m
3659:e
3646:,
3641:2
3637:e
3633:,
3628:1
3624:e
3620:{
3617:=
3614:E
3594:}
3589:n
3585:v
3578:,
3569:,
3564:2
3560:v
3556:,
3551:1
3547:v
3543:{
3540:=
3537:V
3492:G
3484:H
3480:G
3472:G
3468:H
3464:G
3457:H
3453:H
3445:H
3441:G
3421:.
3418:H
3415:=
3405:)
3396:H
3392:(
3360:.
3357:}
3352:i
3348:e
3339:m
3335:x
3326:i
3322:e
3318:{
3315:=
3310:m
3306:X
3282:}
3277:m
3273:X
3269:{
3249:}
3244:i
3240:e
3236:{
3216:H
3190:H
3163:.
3159:)
3155:}
3152:A
3144:i
3140:e
3136:,
3131:e
3127:I
3120:i
3112:i
3108:e
3104:{
3101:,
3098:A
3094:(
3090:=
3087:A
3081:H
3054:X
3048:A
3025:.
3021:)
3017:}
3014:J
3008:i
3000:i
2996:e
2992:{
2989:,
2986:X
2982:(
2958:J
2936:e
2932:I
2925:J
2909:.
2897:}
2894:)
2887:A
2880:A
2877:(
2871:e
2865:E
2859:e
2856:{
2853:=
2846:E
2825:A
2819:e
2814:E
2808:e
2800:=
2793:A
2772:)
2765:E
2761:,
2754:A
2747:A
2744:(
2741:=
2738:)
2733:A
2729:H
2725:(
2722:x
2719:E
2696:)
2691:A
2687:H
2683:(
2680:x
2677:E
2655:A
2651:H
2630:H
2612:A
2608:H
2589:.
2585:)
2581:}
2572:A
2566:e
2563:,
2560:E
2554:e
2548:A
2542:e
2539:{
2536:,
2533:A
2529:(
2525:=
2520:A
2516:H
2492:X
2486:A
2464:A
2460:H
2426:e
2422:I
2399:v
2395:I
2371:,
2368:}
2357:i
2353:e
2349:,
2346:X
2338:i
2334:e
2330:,
2325:e
2321:I
2314:i
2306:i
2302:e
2298:{
2295:=
2292:E
2263:,
2260:}
2255:v
2251:I
2244:i
2236:i
2232:x
2228:{
2225:=
2222:X
2199:)
2196:E
2193:,
2190:X
2187:(
2184:=
2181:H
2149:.
2137:.
2087:k
2066:2
2060:k
2038:k
2014:k
1992:k
1969:k
1947:k
1934:.
1911:V
1907:U
1887:d
1867:d
1845:d
1826:)
1788:k
1770:}
1764:,
1761:.
1758:.
1755:.
1752:,
1749:3
1746:,
1743:2
1740:,
1737:1
1734:{
1703:n
1696:n
1642:.
1630:;
1624:;
1612:;
1596:;
1564:k
1560:k
1415:)
1412:E
1409:,
1406:X
1403:(
1376:|
1372:e
1368:|
1347:X
1341:D
1321:X
1315:C
1288:)
1284:|
1280:C
1276:|
1272:,
1268:|
1264:D
1260:|
1256:(
1253:=
1249:|
1245:e
1241:|
1220:)
1217:C
1214:,
1211:D
1208:(
1205:=
1202:e
1179:E
1155:X
1135:)
1132:E
1129:,
1126:X
1123:(
1087:C
1059:D
1029:E
1023:)
1020:C
1017:,
1014:D
1011:(
991:X
971:E
935:X
915:)
912:E
909:,
906:X
903:(
852:}
849:)
846:}
843:6
840:{
837:,
834:}
831:5
828:,
825:3
822:{
819:(
800:,
797:)
794:}
791:5
788:,
785:4
782:{
779:,
776:}
773:3
770:,
767:2
764:{
761:(
742:,
739:)
736:}
733:1
730:{
727:,
724:}
721:3
718:{
715:(
696:,
693:)
690:}
687:3
684:{
681:,
678:}
675:2
672:{
669:(
650:,
647:)
644:}
641:2
638:{
635:,
632:}
629:1
626:{
623:(
620:{
601:=
598:}
593:5
589:a
585:,
580:4
576:a
572:,
567:3
563:a
559:,
554:2
550:a
546:,
541:1
537:a
533:{
530:=
527:E
507:}
504:6
501:,
498:5
495:,
492:4
489:,
486:3
483:,
480:2
477:,
474:1
471:{
468:=
465:X
428:}
425:}
420:4
416:v
412:{
393:,
390:}
385:6
381:v
377:,
372:5
368:v
364:,
359:3
355:v
351:{
332:,
329:}
324:3
320:v
316:,
311:2
307:v
303:{
284:,
281:}
276:3
272:v
268:,
263:2
259:v
255:,
250:1
246:v
242:{
239:{
220:=
217:}
212:4
208:e
204:,
199:3
195:e
191:,
186:2
182:e
178:,
173:1
169:e
165:{
162:=
159:E
139:}
134:7
130:v
126:,
121:6
117:v
113:,
108:5
104:v
100:,
95:4
91:v
87:,
82:3
78:v
74:,
69:2
65:v
61:,
56:1
52:v
48:{
45:=
42:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
