Characterization of Graphs with Villainy 2

Let $f$ be an optimal proper coloring of a graph $G$ and let $c$ be a coloring of the vertices of $G$ obtained by permuting the colors on vertices in the proper coloring $f$. The villainy of $c$, written $B(c)$, is the minimum number of vertices that must be recolored to obtain a proper coloring of $G$ with the additional condition that the number of times each color is used does not change. The villainy of $G$ is defined as $B(G)=max_{c}B(c)$, over all optimal proper colorings of $G$. In this paper, we characterize graphs $G$ with $B(G)=2$.