Geodesic
The periphery graph of a median graph
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 17-32

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The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join of two graphs and show that they are precisely Cartesian products of median graphs. Path-like median graphs are introduced as the graphs whose periphery graph has independence number 2, and it is proved that there are path-like median graphs with arbitrarily large geodetic number. Peripheral expansion with respect to periphery graph is also considered, and connections with the concept of crossing graph are established.
Keywords: median graph, Cartesian product, geodesic, periphery, peripheral expansion 
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Brešar, Boštjan; Changat, Manoj; Subhamathi, Ajitha; Tepeh, Aleksandra. The periphery graph of a median graph. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 17-32. http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a1/