Geodesic
A characterization of planar median graphs
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 41-48

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Median graphs have many interesting properties. One of them is-in connection with triangle free graphs-the recognition complexity. In general the complexity is not very fast, but if we restrict to the planar case the recognition complexity becomes linear. Despite this fact, there is no characterization of planar median graphs in the literature. Here an additional condition is introduced for the convex expansion procedure that characterizes planar median graphs.
Keywords: median graphs, planar graphs, expansion 
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Peterin, Iztok. A characterization of planar median graphs. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 41-48. http://geodesic.mathdoc.fr/item/DMGT_2006_26_1_a3/