Finding and counting small induced subgraphs efficiently

We give two algorithms for listing all simplicial vertices of a gragh running in rime O(n(alpha)) and O(e(2 alpha/(alpha+1))) = O(e(1.41)) respectively, where n and e denote the number of vertices and edges in the graph and O(n(alpha)) is the time needed to perform a fast matrix multiplication. We present new algorithms for the recognition of diamond-free graphs (O(n(alpha) + e(3/2))), claw-free graphs (O(e((alpha+1)/2)) = O(e(1.69))), and K-4-free graphs (O(e((alpha+1)/2)) = O(e(1.69))). Furthermore, we show that counting the number of K-4's in a graph can be done in time O(e((alpha+1)/2)). For all other graphs on four vertices we can count within O(n(alpha) + e(1.69)) time the number of occurrences as induced subgraph. (C) 2000 Elsevier Science B.V. All rights reserved.