Counting 4×4 matrix partitions of graphs

Given a symmetric matrix M ∈ {0, 1, ∗}D×D, an M-partition of a graph G is a function from V (G) to D such that no edge of G is mapped to a 0 of M and no non-edge to a 1. We give a computer-assisted proof that, when |D| = 4, the problem of counting the M-partitions of an input graph is either in FP or is #P-complete. Tractability is proved by reduction to the related problem of counting list M-partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small number of matrices, which we resolve manually to establish the dichotomy. We conjecture that the dichotomy also holds for |D| > 4. More specifically, we conjecture that, for any symmetric matrix M ∈ {0, 1, ∗}D×D, the complexity of counting M-partitions is the same as the related problem of counting list M-partitions.