Geodesic
Path-Neighborhood Graphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 731-745.

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A path-neighborhood graph is a connected graph in which every neighborhood induces a path. In the main results the 3-sun-free path-neighborhood graphs are characterized. The 3-sun is obtained from a 6-cycle by adding three chords between the three pairs of vertices at distance 2. A P_k-graph is a path-neighborhood graph in which every neighborhood is a P_k, where P_k is the path on k vertices. The P_k-graphs are characterized for k ≤ 4.
Keywords: path-neighborhood graph, outerplanar graph, MOP, snake, 3- sun, k-fun 
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Laskar, R.C.; Mulder, Henry Martyn. Path-Neighborhood Graphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 731-745. http://geodesic.mathdoc.fr/item/DMGT_2013_33_4_a8/

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