Graph homomorphisms with complex values: A dichotomy theorem

Abstract

Graph homomorphism problem has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function Z A (G) is defined as where G=(V, E) is any undirected graph. The function Z A (G) can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of Z A (G), for arbitrary complex valued symmetric matrices A. Building on the work by Dyer and Greenhill [1], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley [3], we prove a complete dichotomy theorem for this problem. © 2010 Springer-Verlag Berlin Heidelberg.