A graph $G$ is called an {\it $M_r(k)$-graph} if $G$ has no $k$-list assignment to its vertices with exactly $r$ vertex colorings. We characterize all $M_3(2)$-graphs. \textbf{In this paper} it is shown that a connected graph $G$ is an $M_3(2)$-graph if and only if each block of $G$ is a complete graph with at least three vertices.
