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Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V (G) is a p-dominating set of the graph G, if every vertex v ∈ V (G)−D is adjacent with at least p vertices of D. The p-domination number γp(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). If G is a nontrivial connected block graph, then we show that γ2(G) ≥ γ(G) + 1, and we characterize all connected block graphs with γ2(G) = γ(G) + 1. Our results generalize those of Volkmann [12] for trees.

Characterization of block graphs with equal 2-domination number and domination number plus one