A tight relation between series-parallel graphs and Bipartite Distance Hereditary graphs

TL;DR

This paper reveals a fundamental connection between bipartite distance hereditary graphs and series-parallel graphs, showing they share the same inductive construction when viewed through the lens of graphic matroids, leading to various applications.

Contribution

It establishes that bipartite distance hereditary graphs and series-parallel graphs are structurally equivalent via their inductive constructions, unifying two well-studied graph classes.

Findings

The two graph classes are the same when viewed as fundamental graphs of a graphic matroid.

Re-proves known properties of bipartite distance hereditary and series-parallel graphs.

Provides new polynomially-solvable instances for the maximum multi-commodity flow problem.

Abstract

Bandelt and Mulder's structural characterization of Bipartite Distance Hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pending vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin's structural characterization of 2-connected series-parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we prove that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series-parallel graphs, to characterize self-dual outer-planar graphs and, finally, to provide a new class of polynomially-solvable instances for the integer multi commodity flow of maximum value.