Matrix Partitions with Finitely Many Obstructions

Each m by m symmetric matrix M over 0, 1, ∗, defines a partition problem, in which an input graph G is to be partitioned into m parts with adjacencies governed by M, in the sense that two distinct vertices in (possibly equal) parts i and j are adjacent if M(i, j) = 1, and nonadjacent if M(i, j) = 0. (The entry ∗ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix S never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without S of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without S which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the...

• Type: Journal
• Year: 2007
• Where: COMBINATORICS
• Authors: Tomás Feder, Pavol Hell, Wing Xie
• Keywords: Characterizations, COMBINATORICS 2007, Matrix, Partition Problems