We study the Chva´tal rank of two linear relaxations of the stable set polytope, the edge constraint and the clique constraint stable set polytope. For some classes of graphs whose stable set polytope is given by 0/1valued constraints only, we present either the exact value of the Chva´tal rank or upper bounds (of the order of their largest cliques and stable sets), which improve the bounds previously known from the literature (of the order of the graph itself).
Abstract. The existence of a densely knit core surrounded by a loosely connected periphery is a common macro-structural feature of social networks. Formally, the CorePeriphery problem is to partition the nodes of an undirected graph G = (V, E) such that a subset X ⊂ V , the core, induces a dense subgraph, and its complement V \X, the periphery, induces a sparse subgraph. Split graphs represent the ideal case in which the core induces a clique and the periphery forms an independent set. The number of missing and superfluous edges in the core and the periphery, respectively, can be minimized in linear time via edit distance to the closest split graph.We show that the CorePeriphery becomes intractable for standard notions of density other than the absolute number of misclassified pairs. Our main tool is a regularization procedure that transforms a given graph with maximum degree d into a d-regular graph with the same clique number by adding at most d · n new nodes. This is of independent interest because it implies that finding a maximum clique in a regular graph is NP-hard to approximate to within a factor of n 1/2−ε for all ε > 0.
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