Vertices of given degree in series-parallel graphs

We show that the number of vertices of a given degree k in several kinds of series-parallel labelled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labelled planar graphs. The proof method is based on generating functions and singularity analysis. In particular we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions. 1. Statement of main results A graph is series-parallel if it does not contain the complete graph K4 as a minor; equivalently, if it does not contain a subdivision of K4. Since both K5 and K3,3 contain a subdivision of K4, by Kuratowski’s theorem a series-parallel graph is planar. An outerplanar graph is a planar graph that can be embedded in the plane so that all vertices are incident to the outer face. They are characterized as those graphs not containing a minor isomorphic to (or a subdivision of) either K4 or K2,3. These are important subfamilies of planar graphs, as they are much simpler but often they already capture the essential structural properties of planar graphs. In particular, they are used as a natural first benchmark for many algorithmic problems and conjectures related to planar graphs The purpose of this paper is to study the number of vertices of given degree in certain classes of labelled planar graphs. In particular, we study labelled outerplanar graphs and series-parallel graphs; in what follows, all graphs are labelled. In order to state our results we introduce the notion of the degree distribution of a random outerplanar graph (the definition for series-parallel graphs is exactly the same). For every n we consider the class of all vertex labelled outerplanar graphs with n vertices. Let Dn denote the degree of a randomly chosen vertex in this class of graphs. Then we say that this class of graphs has a degree distribution if there exist non-negative numbers dk with ∑ k≥0 dk = 1 such that for all k lim n→∞ Pr{Dn = k} = dk. In a companion paper [8], we have established that the classes of 2-connected, connected or all outerplanar graphs, as well as the corresponding classes of seriesparallel graphs have a degree distribution. We describe briefly the degree distribution in the outerplanar case, which is the simplest one, and refer to [8] for the other 1Alternatively we can define Dn as the degree of the vertex with label 1. 1 2 MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY cases. Let dk be defined as before for outerplanar graphs, let D(x) = 1 + x−√1− 6x + x2 4 , and let g(x,w) = xw + xw 2 2D(x)− x 1− (2D(x)− x)w The function D(x) is a minor modification of function A(x) in (2.1), and g(x,w) is the generating function of rooted of 2-connected outerplanar graphs, where w marks the degree of the root. Then we have ∑ k≥0 dkw k = ρ · ∂ ∂x e |x=τ , where ρ and τ are constants defined analytically and having approximate values ρ ≈ 0.1366 and τ ≈ 0.1708. A plot of the distribution is given in Figure 1. For comparison, we have added also a plot of the distribution for 2-connected outerplanar graphs, whose probability generating function can be computed directly from g(x,w). 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35