  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).
