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

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