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In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle 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 clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs or triangulated graphs.
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In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals. Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.
In the mathematical area of graph theory, a chordal bipartite graph is a bipartite graph B = (X,Y,E) in which every cycle of length at least 6 in B has a chord, i.e., an edge that connects two vertices that are a distance > 1 apart from each other in the cycle. A better name would be weakly chordal and bipartite since chordal bipartite graphs are in general not chordal as the induced cycle of length 4 shows.
References
- ↑ Golumbic, Martin Charles; Trenk, Ann N. (2004), Tolerance graphs, Cambridge Studies in Advanced Mathematics, 89, Cambridge University Press, Cambridge, p. xii+265, doi:10.1017/CBO9780511542985, ISBN 0-521-82758-2, MR 2051713
- ↑ Golumbic, Martin C.; Monma, Clyde L. (1982), "A generalization of interval graphs with tolerances", Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982), Congressus Numerantium, 35: 321–331, MR 0725892
- 1 2 "Graphclass: tolerance", Information System on Graph Classes and their Inclusions, retrieved 2019-09-30
- ↑ Golumbic, Martin Charles; Monma, Clyde L.; Trotter, William T. Jr. (1984), "Tolerance graphs", Discrete Applied Mathematics, 9 (2): 157–170, doi:10.1016/0166-218X(84)90016-7, MR 0761599
- ↑ Mertzios, George B.; Sau, Ignasi; Zaks, Shmuel (2011), "The recognition of tolerance and bounded tolerance graphs" (PDF), SIAM Journal on Computing, 40 (5): 1234–1257, doi:10.1137/090780328, MR 2854571