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 whose black vertices are bonds linking together these 3-components, arising from separating pairs of vertices of g. Two fundamental 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 series, in particular the edge index series. Numerous enumerative consequences are discussed.
Cite
@article{arxiv.0712.1869,
title = {Two-connected graphs with prescribed three-connected components},
author = {Andrei Gagarin and Gilbert Labelle and Pierre Leroux and Timothy Walsh},
journal= {arXiv preprint arXiv:0712.1869},
year = {2010}
}
Comments
Work presented at the Ottawa-Carleton Discrete Mathematics Workshop, May 25-26, 2007 and at the Seminaire Lotharingien de Combinatoire, Bertinoro, Italy, September 24-26, 2007. 32 pages. 11 pdf figures. Version 2: Minor revisions, one Table added