Dominating a Family of Graphs with Small Connected Subgraphs

Let F = {G_l,...,G_t} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F,k)-core if, for each υ ∈ V and for each i = 1,...,t, there are at least k neighbours of υ in G_i that belong to D. The subset D is called a connected (F,k)-core if the subgraph induced by D in each G_i is connected. Let δ_i be the minimum degree of G_i and let δ(F) = min^t_i=1 δ_i. Clearly, an (F,k)-core exists if and only if δ(F) ≥ k, and a connected (F,k)-core exists if and only if δ(F) ≥ k and each G_i is connected. Let c(k,F) and c_c(k,F) be the minimum size of an (F,k)-core and a connected (F,k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √ln δ: c(k, F) ≤ nln δ/δ(1 + oδ(1)), c_c(k, F) ≤ nln δ/δ(1 + oδ(1)). The results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.