$ss$-EXCELLENCE IN GRAPHS

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. A subset $S$ of $V(G)$ is called a semi-strong set abbreviated as $ss$-set if $|N[v] \cap S| \leq 1$ for all $v$ in $V(G)$. This concept was introduced by E. Sampathkumar in the paper titled Semi-strong chromatic number of a graph. Any $ss$-set has hereditary property. That is, a subset of an $ss$-set is an $ss$-set. So, an $ss$-set is maximal iff for any $u \in (V-S)$, there exists $v \in V(G)$, $v \neq u$ such that $v$ is adjacent with $u$ and a vertex of $S$. Excellence is studied with respect to several parameters like domination. A vertex $u$ is $\alpha$-good with respect to the parameter $\alpha$ if $u$ belongs to a minimum (maximum) $\alpha$-set of $G$. A graph $G$ is $\alpha$-excellent if every vertex of $G$ is $\alpha$-good. A graph G is $ss$ - excellent if every vertex of G is $ss$ - good. $ss$ - excellence and $ss$ - just excellence are studied in this paper.