Line Graphs with a Sequentially Cohen-Macaulay Clique Complex

Let H be a simple undirected graph and G = L(H) be its line graph. Assume that Ä(G) denotes the clique complex of G. We show that Ä(G) is sequentially Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable. Furthermore, we state a complete characterization of those H for which Ä(G) is sequentially Cohen-Macaulay.