Minimal obstructions for a matrix partition problem in chordal graphs

Abstract

If $M$ is an $m \times m$ matrix over $\{ 0, 1, \ast \}$, an $M$-partition of a graph $G$ is a partition $(V_1, \dots V_m)$ such that $V_i$ is completely adjacent (non-adjacent) to $V_j$ if $M_{ij} = 1$ ($M_{ij} = 0$), and there are no further restrictions between $V_i$ and $V_j$ if $M_{ij} = \ast$. Having an $M$-partition is a hereditary property, thus it can be characterized by a set of minimal obstructions (forbidden induced subgraphs minimal with the property of not having an $M$-partition). It is known that for every $3 \times 3$ matrix $M$ over $\{ 0, 1, \ast \}$, there are finitely many chordal minimal obstructions for the problem of determining whether a graph admits an $M$-partition, except for two matrices, $M_1 = \left( \begin{array}{ccc} 0 & \ast & \ast \\ \ast & 0 & 1 \\ \ast & 1 & 0 \end{array} \right)$ and $M_2 = \left( \begin{array}{ccc} 0 & \ast & \ast \\ \ast & 0 & 1 \\ \ast & 1 & 1 \end{array} \right)$. For these two matrices an infinite family of chordal minimal obstructions is known (the same family for both matrices), but the complete set of minimal obstructions is not. In this work we present the complete family of chordal minimal obstructions for $M_1$.