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 P 5 , or H has two connected components and is an induced subgraph of 2 P 3 . ompletes 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 ℓ P 2 , ℓ copies of a P 2 , if and only if ℓ ⩽ 2 [D. Meister, A complete characterisation of minimal triangulations of 2 K 2 -free graphs, Discrete Mathematics 306 (2006) 3327–3333].