A characterization of domination weak bicritical graphs with large diameter

The domination number of a graph $G$, denoted by $\gamma (G)$, is the minimum cardinality of a dominating set of $G$. A vertex of a graph is called critical if its deletion decreases the domination number, and a graph is called critical if its all vertices are critical. A graph $G$ is called weak bicritical if for every non-critical vertex $x\in V(G)$, $G-x$ is a critical graph with $\gamma (G-x)=\gamma (G)$. In this paper, we characterize the connected weak bicritical graphs $G$ whose diameter is exactly $2\gamma (G)-2$. This is a generalization of some known results concerning the diameter of graphs with a domination-criticality.