  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 $M_1$.
