559:
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A cycle (black) with two chords (green). As for this part, the graph is chordal. However, removing one green edge would result in a non-chordal graph. Indeed, the other green edge with three black edges would form a cycle of length four with no
184:
and the second consisting of the non-neighbors. When this splitting process has been performed for all vertices, the sequence of sets has one vertex per set, in the reverse of a perfect elimination ordering.
168:. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The algorithm repeatedly chooses a vertex
71:
in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a
425:
585:
that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. Gavril showed that the subtree graphs are exactly the chordal graphs.
447:
is a set of vertices the removal of which leaves the remaining graph disconnected; a separator is minimal if it has no proper subset that is also a separator. According to a theorem of
850:-trees are their own chordal completions, and form a subclass of the chordal graphs. Chordal completions can also be used to characterize several other related classes of graphs.
1180:
546:. That is, they are the graphs that have a recursive decomposition by clique separators into smaller subgraphs. For this reason, chordal graphs have also sometimes been called
526:
496:
1417:
Kaplan, Haim; Shamir, Ron; Tarjan, Robert (1999), "Tractability of
Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs",
188:
Since both this lexicographic breadth first search process and the process of testing whether an ordering is a perfect elimination ordering can be performed in
215:
use this connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.
451:, chordal graphs are graphs in which each minimal separator is a clique; Dirac used this characterization to prove that chordal graphs are
1390:"Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition, and consecutive ones testing"
777:
is a triangle, because peripheral cycles are a special case of induced cycles. Strangulated graphs are graphs that can be formed by
1704:
672:
are another subclass of
Ptolemaic graphs in which every two maximal cliques have at most one vertex in common. A special type is
458:
The family of chordal graphs may be defined inductively as the graphs whose vertices can be divided into three nonempty subsets
1394:
1220:
239:. To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each vertex
1619:
1139:
17:
842:
are the graphs to which no additional edges can be added without increasing their treewidth to a number larger than
341:
1469:
Parra, Andreas; Scheffler, Petra (1997), "Characterizations and algorithmic applications of chordal graph embeddings",
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The largest maximal clique is a maximum clique, and, as chordal graphs are perfect, the size of this clique equals the
165:
1172:
1459:
164:) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as
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963:
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A chordal graph with eight vertices, represented as the intersection graph of eight subtrees of a six-node tree.
1580:
235:, while non-chordal graphs may have exponentially many. This implies that the class of chordal graphs has
1324:
135:
1281:
Fomin, Fedor V.; Villanger, Yngve (2013), "Subexponential
Parameterized Algorithm for Minimum Fill-In",
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equal to one less than the size of the largest clique in the graph; the tree decomposition of any graph
1439:
1350:
Gavril, Fănică (1974), "The intersection graphs of subtrees in trees are exactly the chordal graphs",
964:"Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations"
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are chordal graphs in which all maximal cliques and all maximal clique separators have the same size.
172:
from the earliest set in the sequence that contains previously unchosen vertices, and splits each set
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as a subgraph of a chordal graph. The tree decomposition of a graph is also the junction tree of the
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in the perfect elimination ordering, and test whether each of the resulting cliques is maximal.
1614:
679:
193:
762:. Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see
1137:; Lipshteyn, Marina (2007), "Recognizing chordal probe graphs and cycle-bicolorable graphs",
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Bender, E. A.; Richmond, L. B.; Wormald, N. C. (1985), "Almost all chordal graphs split",
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that is not part of the cycle but connects two vertices of the cycle. Equivalently, every
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divides the polynomial, as it should.) Clearly, this computation depends on chordality.
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The set of all perfect elimination orderings of a chordal graph can be modeled as the
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1538:
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1366:
1273:
1213:"Enumerating and characterizing the perfect elimination orderings of a chordal graph"
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Rose, Donald J. (December 1970), "Triangulated graphs and the elimination process",
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of chordal graphs and maximal planar graphs. Therefore, strangulated graphs include
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of a chordal graph in polynomial-time, while the same problem for general graphs is
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of the sequence into two smaller subsets, the first consisting of the neighbors of
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whereas the probe graph problem on chordal graphs has polynomial-time complexity.
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in a graph is an ordering of the vertices of the graph such that, for each vertex
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628:, a special case of trees. Therefore, they are a subfamily of chordal graphs.
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Proc. of 19th
International Colloquium on Automata Languages and Programming
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of a chordal graph is easy to compute. Find a perfect elimination ordering
277:
algorithm to the vertices in the reverse of a perfect elimination ordering.
1598:
1446:, CMS Books in Mathematics, vol. 11, Springer-Verlag, pp. 65–84,
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Habib, Michel; McConnell, Ross; Paul, Christophe; Viennot, Laurent (2000),
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A representation of a chordal graph as an intersection of subtrees forms a
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Another application of perfect elimination orderings is finding a maximum
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99:
40:
823:, and moreover, is solvable in parameterized subexponential time. The
102:, and several problems that are hard on other classes of graphs such as
1257:
778:
750:. Other superclasses of chordal graphs include weakly chordal graphs,
625:
1641:
1617:; Bornstein, C.F. (1994), "Clique graphs of chordal and path graphs",
1304:
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1184:, Lecture Notes in Computer Science, vol. 623, pp. 273–283,
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558:
107:
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1438:
Maffray, Frédéric (2003), "On the coloration of perfect graphs", in
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894:
192:, it is possible to recognize chordal graphs in linear time. The
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of a chordal completion chosen to minimize this clique size. The
676:, where the common vertex is the same for every pair of cliques.
665:
117:
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may be solved in polynomial time when the input is chordal. The
30:
1549:(1976), "Algorithmic aspects of vertex elimination on graphs",
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as a subgraph. The parameterized version of minimum fill-in is
720:
1583:; Weaver, R. W. (1984), "A generalization of chordal graphs",
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231:. More generally, a chordal graph can have only linearly many
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of an arbitrary graph may be characterized by the size of the
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are a subclass of
Ptolemaic graphs that are both chordal and
1112:
Berge, Claude (1967), "Some
Classes of Perfect Graphs", in
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are a subclass of 2-trees, and therefore are also chordal.
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From a collection of subtrees of a tree, one can define a
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An alternative characterization of chordal graphs, due to
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87:: a chordal completion of a graph is typically called a
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of subtrees of a tree. They are sometimes also called
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650:-vertex chordal graphs that are split approaches one.
1080:
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273:: an optimal coloring may be obtained by applying a
1048:"On chordal graphs and their chromatic polynomials"
420:{\displaystyle (x-N_{1})(x-N_{2})\cdots (x-N_{n}).}
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1527:Journal of Mathematical Analysis and Applications
600:can be viewed in this way as a representation of
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746:Chordal graphs are a subclass of the well known
709:degree-two vertices, adjacent to the edges of a
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1444:Recent Advances in Algorithms and Combinatorics
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831:is one less than the number of vertices in a
118:Perfect elimination and efficient recognition
1657:Information System on Graph Class Inclusions
1380:Algorithmic Graph Theory and Perfect Graphs
952:, Remark 2.5, calls this method well known.
682:are graphs that are chordal and contain no
624:are the intersection graphs of subtrees of
1500:Patil, H. P. (1986), "On the structure of
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734:, or equivalently planar 3-trees. Maximal
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634:are graphs that are both chordal and the
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269:of the chordal graph. Chordal graphs are
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1325:"Incidence matrices and interval graphs"
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29:
27:Graph where all long cycles have a chord
1437:
1173:"Two strikes against perfect phylogeny"
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901:Bodlaender, Fellows & Warnow (1992)
154:it has a perfect elimination ordering.
114:in the chordal graphs that contain it.
14:
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913:Berry, Golumbic & Lipshteyn (2007)
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656:are graphs that are both chordal and
640:Bender, Richmond & Wormald (1985)
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1620:SIAM Journal on Discrete Mathematics
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1140:SIAM Journal on Discrete Mathematics
1120:, Academic Press, pp. 155–165,
1118:Graph Theory and Theoretical Physics
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870:
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1244:(1961), "On rigid circuit graphs",
815:) is a chordal graph that contains
338:. The chromatic polynomial equals
94:Chordal graphs are a subset of the
24:
1007:Kaplan, Shamir & Tarjan (1999)
925:Szwarcfiter & Bornstein (1994)
693:) as an induced subgraph. Here an
646:goes to infinity, the fraction of
219:Maximal cliques and graph coloring
166:lexicographic breadth-first search
25:
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789:Chordal completions and treewidth
328:in that ordering. For instance,
314:equal the number of neighbors of
158:Rose, Lueker & Tarjan (1976)
1353:Journal of Combinatorial Theory
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612:Relation to other graph classes
554:Intersection graphs of subtrees
243:together with the neighbors of
1705:Intersection classes of graphs
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705:together with a collection of
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1234:10.1016/S0304-3975(03)00221-4
1207:Chandran, L. S.; Ibarra, L.;
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642:showed that, in the limit as
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1539:10.1016/0022-247x(70)90282-9
1471:Discrete Applied Mathematics
1442:; Sales, Cláudia L. (eds.),
1395:Theoretical Computer Science
1367:10.1016/0095-8956(74)90094-X
1221:Theoretical Computer Science
1031:Parra & Scheffler (1997)
1019:Fomin & Villanger (2013)
124:perfect elimination ordering
98:. They may be recognized in
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995:Seymour & Weaver (1984)
427:(The last factor is simply
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1545:Rose, D.; Lueker, George;
1065:10.7146/math.scand.a-14421
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258:of chordal graphs are the
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1552:SIAM Journal on Computing
1431:10.1137/S0097539796303044
1096:10.1017/S1446788700023077
821:fixed parameter tractable
803:is an arbitrary graph, a
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769:Every chordal graph is a
1376:Golumbic, Martin Charles
1190:10.1007/3-540-55719-9_80
1135:Golumbic, Martin Charles
1052:Mathematica Scandinavica
1046:Agnarsson, Geir (2003),
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754:, odd-hole-free graphs,
1586:Journal of Graph Theory
1452:10.1007/0-387-22444-0_3
1342:10.2140/pjm.1965.15.835
1323:; Gross, O. A. (1965),
680:Strongly chordal graphs
606:junction tree algorithm
521:{\displaystyle S\cup B}
491:{\displaystyle A\cup S}
1599:10.1002/jgt.3190080206
1083:J. Austral. Math. Soc.
701:-vertex chordal graph
662:Quasi-threshold graphs
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213:Chandran et al. (2003)
194:graph sandwich problem
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1211:; Sawada, J. (2003),
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260:dually chordal graphs
196:on chordal graphs is
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282:chromatic polynomial
247:that are later than
146:in the order form a
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51:is one in which all
18:Chord (graph theory)
728:Apollonian networks
658:distance hereditary
638:of chordal graphs.
592:of the graph, with
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271:perfectly orderable
85:triangulated graphs
77:intersection graphs
1668:Weisstein, Eric W.
1258:10.1007/BF02992776
805:chordal completion
795:Chordal completion
771:strangulated graph
766:in graph theory).
736:outerplanar graphs
590:tree decomposition
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1547:Tarjan, Robert E.
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846:. Therefore, the
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779:clique-sums
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636:complements
626:path graphs
570:, involves
237:few cliques
229:NP-complete
209:antimatroid
205:basic words
198:NP-complete
190:linear time
100:linear time
1689:Categories
1642:11422/1497
1209:Ruskey, F.
1199:1874/16653
1040:References
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617:Subclasses
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