The reconstruction of separable graphs with 2-connected trunks that are series-parallel networks

Abstract

A 2-connected graph has a cleavage unit-virtual edge decomposition which is due to Tutte [8]. Cleavage units are either polygons, bonds (planar duals to polygons) or 3-connected simple graphs. When all cleavage units are polygons or bonds, such graphs are called series-parallel networks. The Ulam graph reconstruction conjecture is open for the class of connected graphs containing circuits and/or pendant vertices. Such graphs can be expressed uniquely in terms of a trunk, a connected subgraph without pendant vertices, and a tree-growth, a forest each connected component of which meets the trunk in a unique root vertex. This paper establishes the reconstruction conjecture when the trunk is a series-parallel network and the tree-growth is ‘non-trivial’. This is accomplished by means of Tutte's decomposition of the trunk, and group theoretical techniques first developed in [2]. Use is made of the restricted nature of the automorphism groups of the polygon and bond cleavage units of the trunk.