Double-crossed chords and distance-hereditary graphs

An early characterization of distance-hereditary graphs is that every cycle of length 5 or more has crossing chords. A new, stronger, property is that in every cycle of length 5 or more, some chord has at least two crossing chords. This new property can be characterized by every block being complete multipartite, and also by the vertex sets of cycles of length 5 or more always inducing 3-connected subgraphs. It can also be characterized by forbidding certain induced subgraphs, as well as by requiring certain (not necessarily induced) subgraphs. A second, even stronger property is that, in every cycle of length 5 or more in a distance-hereditary graph, every chord has at least two crossing chords. This second property has characterizations that parallel those of the first property, including by every block being complete bipartite or complete, and also by the vertex sets of cycles of length 5 or more always inducing nonplanar subgraphs.