On a property of minimal triangulations

A graph H has the property MT, if for all graphs G, G is H-free if and only if every minimal (chordal) triangulation of G is H-free. We show that a graph H fulfils property MT if and only if H is edgeless, H is connected and is an induced subgraph of P5, or H has two connected components and is an induced subgraph of 2P3. This completes results of Parra and Scheffler, who have shown that MT holds for H = Pk, the path on k vertices, if and only if k 5 [11], and of Meister, who proved that MT holds for P2, copies of a P2, if and only if 2 [9].