Inequalities of Independence Number, Clique Number and Connectivity of Maximal Connected Domination Critical Graphs

Abstract

A $k$-$γ_{c}$-edge critical graph is a graph $G$ with the connected domination number $γ_{c}(G) = k$ and $γ_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be $k$-$γ_{c}$-vertex critical if $γ_{c}(G) = k$ and $γ_{c}(G - v) < k$ for all $v \in V(G)$. A maximal $k$-$γ_{c}$-vertex critical graph is a graph which are both $k$-$γ_{c}$-edge critical and $k$-$γ_{c}$-vertex critical. Let $κ, δ, ω$ and $α$ be respectively connectivity minimum degree, clique number and independence number. In this paper, we prove that every maximal $3$-$γ_{c}$-vertex critical graph $G$ satisfies $α\leq δ$ and this bound is best possible. We prove further that $G$ satisfies $α+ ω\leq n - 1$ and we also characterize all such graphs achieving the upper bounds. We finally show that if $G$ satisfies $κ< δ$, then every two vertices of $G$ are joined by hamiltonian path.