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Bisimplicial separators
Milanič, Martin, 1980- ...
A minimal separator of a graph G is a set S ⊆ V (G) such that there exist vertices a, b ∈ V (G)⧹S with the property that S separates a from b in G, but no proper subset of S does. For an integer k ≥ ...0, we say that a minimal separator is k‐simplicial if it can be covered by k cliques and denote by Gk the class of all graphs in which each minimal separator is k‐simplicial. We show that for each k ≥ 0, the class Gk 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 Gk. We also give a complete list of minimal forbidden induced minors for G2. Next, we show that, for k ≥ 1, every nonnull graph in Gk has a k‐simplicial vertex, that is, a vertex whose neighborhood is a union of k cliques; we deduce that the MAXIMUM WEIGHT CLIQUE problem can be solved in polynomial time for graphs in G2. Further, we show that, for k ≥ 3, it is NP‐hard to recognize graphs in Gk; the time complexity of recognizing graphs in G2 is unknown. We also show that the MAXIMUM CLIQUE problem is NP‐hard for graphs in G3. Finally, we prove a decomposition theorem for diamond‐free graphs in G2 (where the diamond is the graph obtained from K4 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 G2, and improved running times for the MAXIMUM WEIGHT CLIQUE and MAXIMUM WEIGHT STABLE SET problems for this class of graphs.
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