Bounds on neighborhood total domination in graphs
In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S.Arumugam, C.Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G G is a dominating set S S in G G with the property that the subgraph induced by the open neighborhood of the set S S has no isolated vertex. The neighborhood total domination number, denoted by ? nt ( G ) , is the minimum cardinality of a NTD-set of G G . Every total dominating set is a NTD-set, implying that ? ( G ) ? ? nt ( G ) ? ? t ( G ) , where ?(G) ? ( G ) and ?t(G) ? t ( G ) denote the domination and total domination numbers of G G , respectively. We show that if G G is a connected graph on n?3 n ? 3 vertices, then ? nt ( G ) ? ( n + 1 ) / 2 and we characterize the graphs achieving equality in this bound.
neighborhood total domination total domination domination