Bounds on Neighborhood Total Domination Numberin Graphs

A dominating set D of G is a subset of V(G), such that every vertex in V(G) \ D is adjacent to at least one vertex in D. A neighborhood total dominating set, abbreviated for NTD set D, is a dominating set of G with an extra property: the subgraph induced by the open neighborhood of D, denoted by G[N(D)], has no isolated vertices. The neighborhood total domination number, denoted by γ_nt(G) , is the minimum cardinality of an NTD set in G. A classical result of Vizing relates the size and the domination number of a graph of given order. In this paper, we present a Vizing-like result for γ_nt(G). Some results for γ_nt(G) in terms of other graphic parameters, such as girth, diameter, and degree of G, are also obtained.