THE NONSPLIT TREE DOMINATION NUMBER OF A GRAPH
S. Muthammai, C. Chitiravalli · Journal of Emerging Technologies and Innovative Research · 2019
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). A tree dominating set D of a graph G is a nonsplit tree dominating set (nstd - set) if the induced subgraph V˗ D is connected. The nonsplit tree domination number γnstd(G) of G is the minimum cardinality of a nonsplit tree dominating set. The connectivity κ(G) of G is the minimum number of vertices whose removal results in a disconnected or trivial graph. A partition {V1, V2, V3, … ,Vn} of V(G), in which each Vi is a nstd - set in G is called a nonsplit tree domatic partition of simply nstd - partition of G. The maximum order of a nstd - partition of G is called the nonsplit tree domatic number of G and is denoted by dnstd(G). In this paper, bounds for γnstd(G) and its exact values for some particular classes of graphs and some special graphs are found and an upper bound for the sum of the nonsplit tree domination number and connectivity of a graph and bounds for dnstd(G) and its exact value for some particular classes of graphs are obtained.