Bounds for the 2-domination number of toroidal grid graphs

A k-dominating set for a graph G(V, E) is a set of vertices D⊆ V such that every vertex v∈V\ D is adjacent to at least k vertices in D. The k-domination number of G, denoted by γ k (G), is the cardinality of a smallest k-dominating set of G. Here we establish lower and upper bounds of γ k (C m ×C n ) for k=2. In some cases, these bounds agree so that the exact 2-domination number is obtained.