Counting Subgraphs via Homomorphisms

Abstract

Counting homomorphisms between graphs has applications in variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this work we show a new application of counting graph homomorphisms to the areas of exact and parameterized algorithms. We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Bjorklund, Husfeldt and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color coding based algorithms of Alon, Yuster, and Zwick.