Nonsplit Neighbourhood Tree Domination Number In Connected Graphs

Abstract

: Let G = (V, E) be a connected graph. A subset D of V is called a dominating set of G if N[D] = V. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by g(G). A dominating set D of a graph G is called a tree dominating set (tr - set) if the induced subgraph áDñ is a tree. The tree domination number γ_tr(G) of G is the minimum cardinality of a tree dominating set. A tree dominating set D of a graph G is called a neighbourhood tree dominating set (ntr - set) if the induced subgraph áN(D)ñ is a tree. The neighbourhood tree domination number γ_ntr(G) of G is the minimum cardinality of a neighbourhood tree dominating set. A tree dominating set D of a graph G is called a nonsplit tree dominating set (nstd - set) if the induced subgraph á V - D ñ is connected. The nonsplit tree domination number γ_nstd(G) of G is the minimum cardinality of a nonsplit tree dominating set. A neighbourhood tree dominating set D of G is called a nonsplit neighbourhood tree dominating set, if the induced subgraph áV(G) ‒ Dñ is connected. The nonsplit neighbourhood tree domination number γ_nsntr(G) of G is the minimum cardinality of a nonsplit neighbourhood tree dominating set of G. In this paper, bounds for γ_nsntr(G) and its exact values for some particular classes of graphs and cartesian product of some standard graphs are found.