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.

f'u~o ne'tcl algorithms jinding relative neighborhood graph RNGc(V) /or a set V oj n points are presented. The first algorithm solves this problem /or input points in (R',L,) metric space in time O(na(n,n)) ij the Delaunay triangulation DT(V) is given. This time perjormance is achieved due to attractive and natural application oj FIND-UNION data structure to represent so-called elimination forest oj edges in DT(V). The second algorithm solves the relative neighborhood graph problem in (Rd,Lp), l

Abstract. We present two methods for finding a lowest common ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on n vertices and m edges.The first method is surprisingly natural and solves the all-pairs LCA problem for the input dag on n vertices and m edges in time O(nm). As a corollary, we obtain an O(n 2 )-time algorithm for finding genealogical distances considerably improving the previously known O(n 2.575 ) timebound for this problem.The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. We solve the latter problem and hence also the all-pairs LCA problem in time O(n 2+ 1 4−ω ), where ω = 2.376 is the exponent of the fastest known matrix multiplication algorithm. This improves the previously known O(n w+3 2 ) time-bound for the general all-pairs LCA problem in dags.

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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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