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.