Just excellence and very excellence in graphs with respect to strong domination

Abstract

A graph $G$ is said to be excellent with respect to strong domination if each $u\in V(G)$, belongs to some $\gamma_s$-set of $G$. $G$ is said to be just excellent with respect to strong domination if each $u\in V(G)$ is contained in a unique $\gamma_s$-set of $G$. A graph $G$ which is excellent with respect to strong domination is said to be very excellent with respect to strong domination if there is a $\gamma_s$-set $D$ of $G$ such that to each vertex $u\in V-D$, there exists a vertex $v\in D$ such that $(D-\{v\})\cup\{u\}$ is a $\gamma_s$-set of $G$. In this paper we study these two classes of graphs. A strong very excellent graph is said to be rigid very excellent with respect to strong domination if the following condition is satisfied. Let $D$ be a very excellent $\gamma_s$-set of $G$. To each $u\not\in D$, let $E(u, D)=\{v\in D: (D-\{v\})\cup\{u\}$ is a $\gamma_s$-set of $G\}$. If $|E(u, D)|=1$ for all $u\not\in D$ then $D$ is said to be a rigid very excellent $\gamma_s$-set of $G$. If $G$ has at least one rigid very excellent $\gamma_s$-set of $G$ then $G$ is said to be a rigid very excellent graph with respect to strong domination (or) a strong rigid very excellent graph. Some results regarding strong very excellent graphs are obtained.