Characterization of unicyclic graphs with equal 2-domination number and domination number plus one - Hansberg, Adriana; Volkmann, Lutz (Utilitas Mathematica, 2008)
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 to 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 connected unicyclic graph, then we show that either γ2(G) ≥ γ(G) + 1 or G = C4, and we characterize all connected unicyclic graphs with γ2(G) = γ(G) + 1.