Semitotal Domination in Claw-Free Cubic Graphs

In this paper, we continue the study of semitotal domination in graphs in [Discrete Math. 324, 13–18 (2014)]. A set S of vertices in G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γ_{t 2}(G) , is the minimum cardinality of a semitotal dominating set of G. This domination parameter is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G) , and the total domination number, γ_t(G). We observe that γ(G) ≤ γ_{t 2}(G) ≤ γ_t(G). A claw-free graph is a graph that does not contain K1,3 as an induced subgraph. We prove that if G is a connected, claw-free, cubic graph of order n≥ 10 , then γ_{t 2}(G) ≤ 4 n/ 11.