Total domination value in graphs

Abstract

A set D C V(G) is a total dominating set of G if for every vertex v ∈ V(G) there exists a vertex u ∞ D such that u and v are adjacent. A total dominating set of G of minimum cardinality is called a γ_t(G)-set. For each vertex v ∞ V(G), we define the total domination value of v, TDV(v), to be the number of γ_t(G)-sets to which v belongs. This definition gives rise to o local study of total domination in graphs. In this paper, we study some basic properties of the TDV function; also, we derive explicit formulas for the TDV of any complete n-partite graph, any cycle, and any path.