Independence - domination duality

Given a system G = (G1,G2, . . . ,Gm) of m graphs on the same vertex set V , define the “joint independence number” α∩(G) as the maximal size of a set which is independent in all graphs Gi . Let also γ∪(G) be the “collective domination number” of the system, which is the minimal number of neighborhoods, each taken from any of the graphs Gi , whose union is V . König’s classical duality theorem can be stated as saying that if m= 2 and both graphs G1,G2 are unions of disjoint cliques then α∩(G1,G2)= γ∪(G1,G2). We prove that a fractional relaxation of α∩, denoted by α∗ ∩, satisfies the condition α∗ ∩(G1,G2) γ∪(G1,G2) for any two graphs G1,G2, and α ∗ ∩(G1,G2, . . . ,Gm) > 2 mγ∪(G1,G2, . . . ,Gm) for any m > 2 and all graphs G1,G2, . . . ,Gm. We prove that the convex hull of the (characteristic vectors of the) independent sets of a graph contains the anti-blocker of the convex hull of the non-punctured neighborhoods of the graph and vice versa. This, in turn, yields α∗ ∩(G1,G2, . . . ,Gm) γ ∗ ∪(G1,G2, . . . ,Gm) as well as a dual result. All these results have extensions to general simplicial complexes, the graphical results being obtained from the special case of the complexes of independent sets of graphs. © 2008 Elsevier Inc. All rights reserved.