Facets for node-capacitated multicut polytopes from path-block cycles with two common nodes

Path-block-cycle inequalities are valid, and sometimes facet-defining, inequalities for polytopes in connection with graph partitioning problems and corresponding multicut problems. Special cases of the inequalities were introduced by De Souza & Laurent (1995) and shown to be facet-defining for the equicut polytope. Generalizations of these inequalities were shown by Ferreira et al. (1996) to be valid for nodecapacitated graph partitioning polytopes on general graphs. This paper considers the special case of the inequalities, where all cycles intersect in two nodes, and establishes conditions under which the associated inequalities induce facets of node-capacitated multicut polytopes and bisection cut polytopes.