For a graph G, let \(\gamma (G)\) and core(G) denote the cardinality of a minimum dominating set of G, and the intersection of all the minimum dominating sets of G, respectively. In this paper, we prove that if G is a \(2K_2\) -free graph with \(\gamma (G)\ge 3\) and without isolated vertices, then \(v\in core(G)\) if and only if \(\gamma (G-v)> \gamma (G);\) moreover, we give an example to answer an open question proposed by Samodivkin whether there is a connected graph G such that \(core(G)\ne \emptyset \) and \(\gamma (G-v)=\gamma (G)\) for each vertex v of G. We also prove that for every \(\{claw,Z_2\}\) -free graph G without isolated vertices, \(v\in core(G)\) if and only if \(\gamma (G-v)> \gamma (G)\) .
