Matrix Partitions of Split Graphs

TL;DR

This paper proves that for split graphs, matrix partition problems have finitely many minimal obstructions, provides bounds on their size, and explores related problems for bipartite and co-bipartite graphs.

Contribution

It establishes the finiteness of minimal obstructions for matrix partition problems on split graphs and provides bounds on their size, extending previous results from cographs.

Findings

Finitely many minimal obstructions for split graphs.

Bounds on the maximum size of minimal split obstructions.

Existence of exponential-sized minimal obstructions.

Abstract

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction. This shows for the first time that some matrices have exponential-sized minimal obstructions of any kind (not necessarily split graphs). We also discuss matrix partitions for bipartite and co-bipartite graphs.