For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $γ(G)$ and $τ(G)$ be the domination number of $G$ and the minimum cardinarity of a vertex cover of $G$, respectively. In this paper, we give a new classification of graphs concerning the domination number of central graphs and minimum vertex covers of graphs. Namely, we show that any graph $G$ with at least three vertices can be classified into one of the two classes of graphs with $γ(C(G))=τ(G)$ and $γ(C(G))=τ(G)+1$, respectively, together with some special properties concerning a vertex cover of $G$. We also give some new results on the domination number of central graphs.