Geodesic
On Some Characterizations of Antipodal Partial Cubes
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 439-453

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We prove that any harmonic partial cube is antipodal, which was conjectured by Fukuda and K. Handa, Antipodal graphs and oriented matroids, Discrete Math. 111 (1993) 245–256. Then we prove that a partial cube G is antipodal if and only if the subgraphs induced by Wab and Wba are isomorphic for every edge ab of G. This gives a positive answer to a question of Klavžar and Kovše, On even and harmonic-even partial cubes, Ars Combin. 93 (2009) 77–86. Finally we prove that the distance-balanced partial cube that are antipodal are those whose pre-hull number is at most 1.
Keywords: diametrical graph, harmonic graph, antipodal graph, distance-balanced graph, partial cube, pre-hull number 
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Polat, Norbert. On Some Characterizations of Antipodal Partial Cubes. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 439-453. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a9/