Finding and counting small induced subgraphs efficiently

Abstract

We give two algorithms for listing all simplicial vertices of a graph running in time O(n α) and O(e 2α/(α+1))= O(e 1.41) , respectively, where n and e denote the number of vertices and edges in the graph and O(n α) is the time needed to perform a fast matrix multiplication. We present new algorithms for the recognition of diamond-free graphs ( O(n α+e 3/2) ), claw-free graphs ( O(e (α+1)/2)= O(e 1.69) ), and K 4 -free graphs ( O(e (α+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 (α+1)/2) . For all other graphs on four vertices we can count within O(n α+e 1.69) time the number of occurrences as induced subgraph.