Combining the Concepts of Residual and Domination in Graphs

Abstract

Let G = (V (G), E(G)) be a simple undirected graph. The domination and average lower domination numbers are vulnerability parameters of a graph. We have investigated a refinement that involves the residual domination and average lower residual domination numbers of these parameters. The lower residual domination number, denoted by ${\gamma }_{v_k}^R(G)$ , is the minimum cardinality of dominating set in G that received from the graph G where the vertex v_k and all links of the vertex v_k are deleted. The residual domination number of graphs G is defined as ${\gamma }^R(G)={\min }_{v_k\in V(G)}\left\{{\gamma }_{v_k}^R(G)\right\}$ . The average lower residual domination number of G is defined by ${\gamma }_{av}^R(G)=\frac{1}{\left|V(G)\right|}{\displaystyle {\sum }_{v_k\in V(G)}{\gamma }_{V_k}^R(G)}$ . In this paper, we define the residual domination and the average lower residual domination numbers of a graph and we present the exact values, upper and lower bounds for some graph families.