Almost All Friendly Matrices Have Many Obstructions

A symmetric m×m matrix M with entries taken from {0, 1, ∗} gives rise to a graph partition problem, asking whether a graph can be partitioned into m vertex sets matched to the rows (and corresponding columns) of M such that, if Mij = 1, then any two vertices between the corresponding vertex sets are joined by an edge, and if Mij = 0 then any two vertices between the corresponding vertex sets are not joined by an edge. The entry ∗ places no restriction on the edges between the corresponding sets. This problem generalises graph colouring and graph homomorphism problems. A graph with no M -partition but such that every proper subgraph does have an M -partition is called a minimal obstruction. Feder, Hell and Xie [5] have defined friendly matrices and shown that non-friendly matrices have infinitely many minimal obstructions. They showed through examples that friendly matrices can have finitely or infinitely many minimal obstructions and gave an example of a friendly matrix with an NP-hard partition problem. Here we show that almost all friendly matrices have infinitely many minimal obstructions and an NP-hard partition problem.