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In graph theory, a well-covered graph is an undirected graph in which the minimal vertex covers all have the same size. Here, a vertex cover is a set of vertices that touches all edges, and it is minimal if removing any vertex from it would leave some edge uncovered. Equivalently, well-covered graphs are the graphs in which all maximal independent sets have equal size. Well-covered graphs were defined and first studied by Michael D. Plummer in 1970.
In graph theory, a skew partition of a graph is a partition of its vertices into two subsets, such that the induced subgraph formed by one of the two subsets is disconnected and the induced subgraph formed by the other subset is the complement of a disconnected graph. Skew partitions play an important role in the theory of perfect graphs.
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), "Chapter 4. Interval probe graphs and sandwich problems", Tolerance Graphs, Cambridge, pp. 63–83.
- 1 2 3 4 Golumbic, Martin Charles; Kaplan, Haim; Shamir, Ron (1995), "Graph sandwich problems", J. Algorithms, 19 (3): 449–473, doi:10.1006/jagm.1995.1047 .
- ↑ Golumbic, Martin Charles (2004), Algorithmic Graph Theory and Perfect Graphs, Annals of Discrete Mathematics, vol. 57 (2nd ed.), Elsevier, p. 279, ISBN 978-0-08-052696-6 .
- ↑ de Figueiredo, C. M. H.; Faria, L.; Klein, S.; Sritharan, R. (2007), "On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs", Theoretical Computer Science , 381 (1–3): 57–67, doi: 10.1016/j.tcs.2007.04.007 , MR 2347393 .
- 1 2 Mahadev, N.V.R.; Peled, Uri N. (1995), Threshold Graphs and Related Topics, Annals of Discrete Mathematics, vol. 57, North-Holland, pp. 19–22, ISBN 978-0-08-054300-0 .
- ↑ Dantas, S.; Klein, S.; Mello, C.P.; Morgana, A. (2009), "The graph sandwich problem for P4-sparse graphs", Discrete Mathematics , 309 (11): 3664–3673, doi: 10.1016/j.disc.2008.01.014 .
- ↑ Dantas, Simone; de Figueiredo, Celina M.H.; Maffray, Frédéric; Teixeira, Rafael B. (2013), "The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem", Discrete Applied Mathematics, 182: 15–24, doi: 10.1016/j.dam.2013.09.004 .