On the 2-Club Polytope of Graphs

Abstract

A k-club is a subset of vertices of a graph that induces a subgraph of diameter at most k, where k is a positive integer. By definition, 1-clubs are cliques and the model is a distance-based relaxation of the clique definition for larger values of k. The k-club model is particularly interesting to study from a polyhedral perspective as the property is not hereditary on induced subgraphs when k is larger than one. This article introduces a new family of facet-defining inequalities for the 2-club polytope that unifies all previously known facets through a less restrictive combinatorial property, namely, independent (distance) 2-domination. The complexity of separation over this new family of inequalities is shown to be NP-hard. An exact formulation of this separation problem and a greedy separation heuristic are also proposed. The polytope described by the new inequalities (and nonnegativity) is then investigated and shown to be integral for acyclic graphs. An additional family of facets is also demonstrated for cycles of length indivisible by three. The effectiveness of these new facets as cutting planes and the difficulty of solving the separation problem in practice are then investigated via computational experiments on a test bed of benchmark instances.

Asymptotic bounds for clustering problems in random graphs

“…It should be noted that for some graph properties, corresponding to other clique relaxations of practical importance, logarithmic bounds do not hold. Consider, for example, an $$$ s $$$‐ club , which is a subset of vertices inducing a subgraph with diameter at most $$$ s $$$ [24, 26, 32]. For $$$ s=1 $$$, $$$ s $$$‐club is a clique, and for $$$ s>1 $$$, any vertex together with its closed neighborhood is an $$$ s $$$‐club.…”