  A graph $G$ is $\alpha$-excellent if every vertex of $G$ is contained in some
maximum independent set of $G$. In this paper, we characterize
$\alpha$-excellent bipartite graphs, $\alpha$-excellent unicyclic graphs,
$\alpha$-excellent simplicial graphs, $\alpha$-excellent chordal graphs,
$\alpha$-excellent block graphs, and we show that every generalized Petersen
graph is $\alpha$-excellent.
