Combinatorial properties and further facets of maximum edge subgraph polytopes

Given a graph G and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset of G such that the number of edges within the subset is maximum. This NP-hard problem arises in the analysis of cohesive subgroups in social networks. In this work we study the polytope P ( G , k ) associated with a straightforward integer programming formulation of the maximum edge subgraph problem. We characterize the graph generated by P ( G , k ) and give a tight bound on its diameter. We give a complete description of P ( K 1 n , k ) , where K 1 n is the star on n + 1 vertices, and we conjecture a complete description of P ( m K 2 , k ) , where m K 2 is the graph composed by m disjoint edges. Finally, we introduce three families of facet-inducing inequalities for P ( G , k ) , which generalize known families of valid inequalities for this polytope.