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We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let H be a fixed graph with k vertices and an independent set of size s. 1. Detecting if an n-vertex graph contains a (not necessarily induced) subgraph isomorphic to H can be done in time O(n ω((k−s)/2 ,1, (k−s)/2)), where ω(p, q, r) is the exponent of fast arithmetic matrix multiplication of an n p × n q matrix by an n q × n r matrix. 2. When s = 2, counting the number of (not necessarily induced) subgraphs isomorphic to H can be done in the same time, i.e., in time O(n ω((k−2)/2 ,1, (k−2)/2)). It follows in particular that we can count the number of subgraphs isomorphic to any H on four vertices that is not K 4 in time O(n ω), where ω = ω(1, 1, 1) is known to be smaller than 2.373. Similarly, we can count the number of subgraphs isomorphic to any H on five vertices that is not K 5 in time O(n ω(2,1,1)), where ω(2, 1, 1) is known to be smaller than 3.257. Finally, we derive inputsensitive variants of our time upper bounds. They are partially expressed in terms of the number m of edges of the input graph and do not rely on fast matrix multiplication.

We consider the following clustering problems: given a general undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP ), or the number of edges between clusters is minimized (Min-ECP ). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

We study the induced subgraph isomorphism problem and the general subgraph isomorphism problem for small pattern graphs. We present a new general method for detecting induced subgraphs of a host graph isomorphic to a fixed pattern graph by reduction to polynomial testing for nonidentity with zero over a field of finite characteristic. It yields new upper time bounds for several pattern graphs on five vertices and provides an alternative combinatorial method for the majority of pattern graphs on four and three vertices. Since our method avoids the large overhead of fast matrix multiplication, it can be of practical interest even for larger pattern graphs. Next, we derive new upper time bounds on counting the number of isomorphisms between a fixed pattern graph with an independent set of size s and a subgraph of the host graph. We also consider a weighted version of the counting problem, when one counts the number of isomorphisms between the pattern graph and lightest subgraphs, providing a slightly slower combinatorial algorithm.

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