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A set of n points in the plane which are not all collinear defines at least ndistinct lines. Chen and Chv\'atal conjectured in 2008 that a similar resultcan be achieved in the broader context of finite metric spaces. This conjectureremains open even for graph metrics. In this article we prove that graphs withno induced house nor induced cycle of length at least~5 verify the desiredproperty. We focus on lines generated by vertices at distance at most 2, definea new notion of ``good pairs'' that might have application in larger families,and finally use a discharging technique to count lines in irreducible graphs.

Graphs with no induced house nor induced hole have the de Bruijn-Erd\H{o}s property