Separator orders in interval, cocomparability, and AT-free graphs

Backer, Jonathan

doi:10.1016/j.dam.2011.01.014

Cited by 1 publication

(2 citation statements)

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“…The polynomial running time of the algorithm given in [30] was based on polynomial-time solvability of the MAXIMUM WEIGHT STABLE SET problem in the class of perfect graphs. Using maximum flow techniques, an improved running time of  n ( ) 4 can be achieved, see [5]. For graphs in  2 , a further improvement can be obtained using LexBFS and recent developments on maximum flow algorithms.…”

Section: Algorithmic Implicationsmentioning

confidence: 99%

“…Minimal separators have been studied since at least the 1960s, when chordal graphs were characterized as precisely those graphs in which all minimal separators are cliques [18]. Minimal separators were subsequently studied in [6] in the context of moplexes, have played an important role in sparse matrix computations via minimal triangulations (for a survey, see [23]), and have also had numerous algorithmic applications (see, e.g., [4,7,10,28]). This paper is a contribution to the study of minimal separators.…”

Section: Introductionmentioning

confidence: 99%

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Bisimplicial separators

Milanič,

Penev,

Pivač

et al. 2024

Journal of Graph Theory

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A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

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Section: Algorithmic Implicationsmentioning

confidence: 99%