Matrix partitions of perfect graphs

Given a symmetric m by m matrix M over 0, 1, ∗, the M -partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i = j) are nonadjacent if M(i, j) = 0, and adjacent if M(i, j) = 1. These matrix partition problems generalize graph colourings and homomorphisms, and arise frequently in the study of perfect graphs; example problems include split graphs, clique and skew cutsets, homogeneous sets, and joins. In this paper we study M -partitions restricted to perfect graphs, and focus on a natural class of simple matrices M with 0, 1 diagonal. For this class of matrices M , all M -partition problems restricted to perfect graphs can be solved in polynomial time, and characterized by a finite set of forbidden induced subgraphs. We identify two distinct patterns of such matrices: For the first pattern we are able to characterize partitionability by small forbidden subgraphs. For the second pattern, we can only guarantee a finite set of forbidden subgraphs (which can be quite large). We also discuss matrices for which partitionability of perfect graphs can not be characterized by finite sets of forbidden subgraphs.