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 satisfies 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 the results of Parra and Scheffler, who have shown that MT holds for H=P_k, the path on k vertices, if and only if k?5 A. Parra, P. Scheffler, Characterizations and algorithmic applications of chordal graph embeddings, Discrete Applied Mathematics 79 (1997) 171-188], and of Meister, who proved that MT holds for ?P₂, ? copies of a P₂, if and only if ??2 D. Meister, A complete characterisation of minimal triangulations of 2K₂-free graphs, Discrete Mathematics 306 (2006) 3327-3333].