  Associated to a simple undirected graph $G$ is a simplicial complex
$\Delta_G$ whose faces correspond to the independent sets of $G$. A graph $G$
is called vertex decomposable if $\Delta_G$ is a vertex decomposable simplicial
complex. We are interested in determining what families of graph have the
property that the complement of $G$, denoted by $\overline{G}$, is vertex
decomposable. We obtain the result that the complement of a connected bipartite
graph is vertex decomposable and so it is Cohen-Macaulay due to pureness of
$\Delta_{\overline{G}}$.
