On the Conjunctive Capacity of Graphs

Abstract

The investigation of the asymptotic behaviour of various graph parameters in powers of a fixed graph G = (V, E) is motivated by problems in information theory and extremal combinatorics. Considering various parameters and/or various notions of graph powers we can arrive at different notions of graph capacities, of which the Shannon capacity is best known. Here we study a related notion of the so-called conjunctive capacity of a graph G, CAND(G), introduced and studied by Gargano, Körner and Vaccaro [5], [6]. To determine CAND(G) is a convex programming problem. In this paper we show that the optimal solution to this problem is unique and describe the structure of the solution in any (simple) graph. We show that its reciprocal value vcC (G) := 1 C AND (G) is an optimal solution of the newly introduced problem of Minimum Capacitary Vertex Cover that is closely related to the LP-relaxation of the Minimum Vertex Cover Problem. We also describe its close connection with the binding number/binding set of a graph, and with the strong crown decomposition of graphs introduced in [2].