We present a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.
We use the framework of graph motif parameters to obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating a parameter f from C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way.
Based on joint work with Holger Dell and Dániel Marx.