Note on inseparability graphs of matroids having exactly one class of orientations

The inseparability graph of an oriented matroid is an invariant of its class of orientations. When an orientable matroid has exactly one class of orientations the inseparability graph of all its orientations is in fact determined by its non-oriented underlying matroid. From this point of view it is natural to ask if inseparability graphs can be used to characterize matroids which have exactly one class of orientations. We give a positive answer to this question in the particular case of series-parallel networks. We prove that series-parallel networks are those orientable matroids for which every orientation has as inseparability graph a complete graph, obtaining, in particular, a simple proof of a theorem of R.J. Duffin. Introducing the notion of labelled inseparability graph of an oriented matroid we characterize labelled inseparability graphs of oriented series-parallel networks and show that the original oriented matroid series-parallel network can be recovered from its labelled inseparability graph. Mathematical Subject Classification (1991): 05B35, 51E20