A Splitting Lemma

Abstract

In this paper, we study the relations between the numerical structure of the optimal solutions of a convex programming problem defined on the edge set of a simple graph and the stability number (i.e. the maximum size of a subset of pairwise non-adjacent vertices) of the graph. Our analysis shows that the stability number of every graph G can be decomposed in the sum of the stability number of a subgraph containing a perfect 2-matching (i.e. a system of vertex-disjoint odd-cycles and edges covering the vertex-set) plus a term computable in polynomial time. As a consequence, it is possible to bound from above and below the stability number in terms of the matching number of a subgraph having a perfect 2-matching and other quantities computable in polynomial time. Our results are closely related to those by Lorentzen, Balinsky, Spielberg, and Pulleyblank on the linear relaxation of the vertex-cover problem. Moreover, The convex programming problem involved has important applications in information theory and extremal set theory where, as a graph capacity formula, has been used to answer a longstanding open question about qualitatively independet sets in the sense of Renyi (L. Gargano, J. K{\"o}rner, and U. Vaccaro, "Sperner capacities", Graphs and combinatorics, 9:31-46, 1993).