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