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Matrix partitions with finitely many obstructions. (English) Zbl 1158.05326

Summary: 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 descriptive and computational complexity of matrix partition problems.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A99 Basic linear algebra
05C75 Structural characterization of families of graphs

Citations:

Zbl 1158.05325
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