AVERAGE TOTAL DOMINATION NUMBER ON GRAPHS

Abstract A network is modelled with the graphs in a situation which the centers are equal to the vertex of graphs and connection lines are equal to the edges of a graph. A graph G is denoted by G = (V (G);E(G)), where V (G) and E(G) are vertex and edge sets of G, respectively. Let v be a vertex in V (G) and p and q be the number of vertices and edges in G. In a graph G = (V (G);E(G)), a subset S ⊆ V (G) of vertices is a dominating set if every vertex in V (G)−S is adjacent to at least one vertex of S. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set. The concept of total domination was introduced by Cockayne, Dawes and Hedetniemi. A dominating set S ⊆ V is a Total Dominating set if the induced subgraph has no isolated vertices. The total domination number γt(G) of a graph G is the minimum cardinality of a total dominating set. Henning introduced the concept of average domination number. The average domination number gag(G) is defined as where gv(G) is the minimum cardinality of a dominating set that contains v. In this paper a new parameter namely average total domination number is defined and is studied for connected graphs. In this paper average total domination number is studied for complete binary tree. Some bounds on average total domination number in terms of total domination number are also established.