Geodesic
Some Variations of Perfect Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 661-668.

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We consider (ψk−γk−1)-perfect graphs, i.e., graphs G for which ψk(H) = γk−1(H) for any induced subgraph H of G, where ψk and γk−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψk−γk−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ2 − γ1)- perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging to this family.
Keywords: k-path vertex cover, distance k-domination number, perfect graphs 
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Dettlaff, Magda; Lemańska, Magdalena; Semanišin, Gabriel; Zuazua, Rita. Some Variations of Perfect Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 661-668. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a10/

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