On domination and annihilation in graphs with claw-free blocks

Let γ(G), ra(G) and ir(G) denote the domination, R-annihilation and irredundance numbers of a graph G, respectively. Graphs whose blocks are claw-free are called script C sign ℱ ℬ-graph. In this paper we establish the best possible upper bounds on the ratios γ(G)/ra(G) and γ(G)/ir(G) in the class of script C sign ℱ ℬ-graphs. The script C sign ℱ ℬ-graphs generalize several classes of graphs for which such ratios have already been investigated. Motivated by our proof methods, we are led to introduce a new family of domination parameters simultaneously generalizing the total domination and k-domination numbers. For two integers, l ≥ 0 and k > 0 a set X of vertices of a graph G=(V, E) is an l-total k-dominating set of G, if every vertex in X has at least l neighbors in X and every vertex in V \ X has at least k neighbors in X. If (at least) one l-total k-dominating set exists, then the l-total k-dominating number γ_l,k(G) is the minimum cardinality of such a set. We prove a best possible upper bound on the ratio γ(G)/γ_1,2(G) in the class of script C sign ℱ ℬ-graphs. Our bounds on γ(G)/ra(G) and γ(G)/ir(G) for a script C sign ℱ ℬ-graph G will follow as an application of this result.