$\mathrm{\Phi }$-Strong(weak) Domination in a Graph

Abstract

Let $G=(V,E)$ be a graph. Let $\mathrm{\Phi }:V\to R$, where $R$ is the set of all reals ($R$ can be replaced by any chain). We say that $u$ $\mathrm{\Phi }$-strongly dominates $v$ and $v$ $\mathrm{\Phi }$-weakly dominates $u$ if $uv\in E$ and $\mathrm{\Phi }(u)\ge \mathrm{\Phi }(v)$. When $\mathrm{\Phi }$ is a constant function, we have the usual domination and when $\mathrm{\Phi }$ is the degree function of the graph, we have the strong (weak) domination studied by Sampathkumar et al. In this paper, we extend the results of O. Ore regarding minimal dominating sets of a graph. We also extend the concept of fully domination balance introduced by Sampathkumar et al and obtain a lower bound for strong domination number of a graph.