Independence-domination duality

Given a system G = (G₁, G₂, ..., G_m) 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 G_i. 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 G_i, 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, ..., G_m) > frac(2, m) γ_∪ (G₁, G₂, ..., G_m) for any m > 2 and all graphs G₁, G₂, ..., G_m. 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, ..., G_m) ≥ γ_∪^* (G₁, G₂, ..., G_m) 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.