Structure and enumeration of two-connected graphs with prescribed three-connected components

Abstract

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc ( g ) associated with any 2-connected graph g, whose white vertices are the 3 -components of g (3-connected components or polygons) and black vertices are bonds arising from separating pairs of vertices of g and linking together the 3-components. Two fundamental species relationships on graphs and networks follow from this construction. The first one is a dissymmetry theorem which leads to the expression of the class B = B ( F ) of 2-connected graphs, all of whose 3-connected components belong to a given class F of 3-connected graphs, in terms of various rootings of B . The second one is a functional equation which characterizes the corresponding class R = R ( F ) of two-pole networks all of whose 3-connected components are in F . All the rootings of B are then expressed in terms of F and R . There follow corresponding identities for all the associated power series, in particular, the edge index series. These results are expressed in terms of species of structures. Using these results we enumerate several classes of labelled and unlabelled graphs, including series-parallel graphs, 2-connected planar graphs, K 3 , 3 -free 2-connected graphs and K 3 , 3 -free projective planar and toroidal graphs.