Planar Graphs

Abstract

At first sight planar graphs and trees have nothing in common despite the trivial fact that trees are planar. Nevertheless there are strong similarities in the combinatorial and asymptotic analysis of trees and planar graphs. Planar graphs contain several (hidden) tree or tree-like structures. The tree-decomposition into cut-vertices and blocks (2-connected components) is the most prominent one. The reduction to 3-connected components is more involved but uses so-called series-parallel networks that are obtained from series-parallel extensions of trees. Thus, it is natural that properly extended tree counting techniques and analytic techniques that have been developed for trees apply. However, these extensions are not straight forward. Several graph theoretical and and also topological concepts have to be combined with combinatorics on trees. There are also different levels of complexity in the asymptotics analysis. From this point of view outerplanar graphs and series-parallel graphs — these are two subclasses of planar graphs that we will study first — are more tree-like than the class of all planar graphs, since the singularity structure of the corresponding generating functions is of square root type $\sqrt {R - x}$, whereas the class of all planar graphs has a dominant singularity of the form (R−x)^3/2. Geometrically this indicates that outerplanar graphs and series-parallel graphs are more or less governed by a one-dimensional topology (as trees) but the class of all planar graphs by a two-dimensional one.