Finding and counting small induced subgraphs efficiently

Abstract

We give two algorithms for listing all simplicial vertices of a graph. The first of these algorithms takes O(nα) time, where n is the number of vertices in the graph and O(nα) is the time needed to perform a fast matrix multiplication. The second algorithm can be implemented to run in (formula presented), where e is the number of edges in the graph. We present a new algorithm for the recognition of diamond-free graphs that can be implemented to run in time O(nα + e3/2). We also present a new recognition algorithm for claw-free graphs. This algorithm can be implemented to run in time (formula presented). It is a fairly easy observation that, within time (formula presented) can be checked whether a graph has a K4. This improves the (formula presented) algorithm mentioned by Alan, Yuster and Zwick. Furthermore, we show that counting the number of K4's in a graph can be done within the same time bound (formula presented). Using the result on the K4's we can count the number of occuren.ces as induced subgraph of any other fixed connected graph on four vertices within O(nα + e1.69).