Geodesic
Dualizing Distance-Hereditary Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 285-296

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Distance-hereditary graphs can be characterized by every cycle of length at least 5 having crossing chords. This makes distance-hereditary graphs susceptible to dualizing, using the common extension of geometric face/vertex planar graph duality to cycle/cutset duality as in abstract matroidal duality. The resulting “DH* graphs” are characterized and then analyzed in terms of connectivity. These results are used in a special case of plane-embedded graphs to justify viewing DH* graphs as the duals of distance-hereditary graphs.
Keywords: distance-hereditary graph, dual graph, graph duality 
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McKee, Terry A. Dualizing Distance-Hereditary Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 285-296. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a17/