an:06535847
Zbl 1329.05232
Henning, Michael A.; Marcon, Alister J.
Vertices contained in all or in no minimum semitotal dominating set of a tree
EN
Discuss. Math., Graph Theory 36, No. 1, 71-93 (2016).
00352066
2016
j
05C69
domination; semitotal domination; trees
Summary: Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, \(\gamma(G)\), and the total domination number, \(\gamma_t(G)\). A set \(S\) of vertices in a graph \(G\) 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, \(\gamma_{t2}(G)\), is the minimum cardinality of a semitotal dominating set of \(G\). We observe that \(\gamma(G) \leq \gamma_{t2}(G) \leq \gamma_t(G)\). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.