Weakly connected domination critical graphs

Abstract

A dominating set $D \subset V(G)$ is a weakly connected dominating set in $G$ if the subgraph $G[D]w = (N{G}[D],E_w)$ weakly induced by $D$ is connected, where $E_{w}$ is the set of all edges with at least one vertex in $D$. The weakly connected domination number $\gamma_w(G)$ of a graph $G$ is the minimum cardinality among all weakly connected dominating sets in $G$. The graph is said to be weakly connected domination critical ($\gamma_w$-critical) if for each $u, v \in V(G)$ with $v$ not adjacent to $u$, $\gamma_w(G + vu) \lt \gamma_w (G)$. Further, $G$ is $k$-$\gamma_w$-critical if $\gamma_w(G)=k$ and for each edge $e \not\in E(G)$, $\gamma_w(G + e) \lt k$. In this paper we consider weakly connected domination critical graphs and give some properties of $3$-$\gamma_w$-critical graphs.