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 γtr(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 γntr(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 γnstd(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 γnsntr(G) of G is the minimum cardinality of a nonsplit neighbourhood tree dominating set of G. In this paper, bounds for γnsntr(G) and its exact values for some particular classes of graphs and cartesian product of some standard graphs are found.