A tight relation between series-parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder's structural characterization of bipartite distance hereditary graphs 16 asserts that such graphs can be built inductively starting from a single vertex and by re- 17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood 18 as an existing one). Dirac and Duffin's structural characterization of 2-connected series- 19 parallel graphs asserts that such graphs can be built inductively starting from a single edge 20 by adding either edges in series or in parallel. In this paper we give an elementary proof 21 that the two constructions are the same construction when bipartite graphs are viewed as 22 the fundamental graphs of a graphic matroid. We then apply the result to re-prove known 23 results concerning bipartite distance hereditary graphs and series-parallel graphs and to 24 provide a new class of polynomially-solvable instances for the integer multi-commodity 25 flow of maximum value.