Representing distance-hereditary graphs with multi-rooted trees

TL;DR

This paper demonstrates that distance-hereditary graphs can be characterized and represented using labelled arboreal networks, extending the tree-based representations used for cographs.

Contribution

It introduces a novel representation of distance-hereditary graphs through labelled arboreal networks, generalizing the tree-based models for cographs.

Findings

Distance-hereditary graphs are exactly those explainable by labelled arboreal networks.

The representation generalizes the known tree-based models for cographs.

Provides a new structural perspective for understanding distance-hereditary graphs.

Abstract

Arboreal networks are a generalization of rooted trees, defined by keeping the tree-like structure, but dropping the requirement for a single root. Just as the class of cographs is precisely the class of undirected graphs that can be explained by a labelled rooted tree (T,t), we show that the class of distance-hereditary graphs is precisely the class of undirected graphs that can be explained by a labelled arboreal network (N,t).