Enumeration and limit laws for series–parallel graphs

Bodirsky, Manuel; Giménez, Omer; Kang, Mihyun; Noy, Marc

doi:10.1016/j.ejc.2007.04.011

Cited by 70 publications

(114 citation statements)

References 8 publications

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“…• i does not interfere with the singularity of C • j ; this is proved also in [3]. As mentioned in the introduction, the equations from Lemma 2.6 linking the B 8.3.…”

Section: -Connectedmentioning

confidence: 77%

“…We now state it for outerplanar graphs but the following lemma is also valid, for example, for series parallel graphs or for general planar graphs [10,3].…”

Section: Connected Outerplanar Graphsmentioning

confidence: 99%

“…Since f (x) = xC (x) satisfies the equation f (x) = e B (f (x)) , the function ψ(u) = ue −B (u) is the functional inverse function of f (x). We know from [3] that there exists ρ 2 = 0.1688 · · · < ρ 1 with ψ (u) = 0. This implies that the radius of convergence of f (x) and consequently that of C (x) is given by ρ = 0.1366 · · · .…”

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confidence: 99%

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Vertices of given degree in series‐parallel graphs

Drmota

Giménez

Noy

2009

Random Struct Algorithms

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Abstract. 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. Statement of main resultsA graph is series-parallel if it does not contain the complete graph K 4 as a minor; equivalently, if it does not contain a subdivision of K 4 . Since both K 5 and K 3,3 contain a subdivision of K 4 , 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 K 4 or K 2,3 . 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 D n denote the degree of a randomly chosen vertex in this class of graphs 1 . Then we say that this class of graphs has a degree distribution if there exist non-negative numbers d k with k≥0 d k = 1 such that for all kIn a companion paper [6], 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 [6] for the other 1 Alternatively we can define Dn as the degree of the vertex with label 1.

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“…• i does not interfere with the singularity of C • j ; this is proved also in [3]. As mentioned in the introduction, the equations from Lemma 2.6 linking the B 8.3.…”

Section: -Connectedmentioning

confidence: 77%

“…We now state it for outerplanar graphs but the following lemma is also valid, for example, for series parallel graphs or for general planar graphs [10,3].…”

Section: Connected Outerplanar Graphsmentioning

confidence: 99%

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Vertices of given degree in series‐parallel graphs

Drmota

Giménez

Noy

2009

Random Struct Algorithms

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“…Next we show that this is the case. Let us remark that in a related problem, counting series-parallel graphs, a very similar situation appears but the analogous ψ function does have a maximum in its domain of definition [2].…”

Section: Asymptotic Estimatesmentioning

confidence: 99%

Asymptotic enumeration and limit laws of planar graphs

Giménez

Noy

2008

J. Amer. Math. Soc.

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137

288

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Abstract. We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate gn ∼ g ·n −7/2 γ n n! for the number gn of labelled planar graphs on n vertices, where γ and g are explicit computable constants. We show that the number of edges in random planar graphs is asymptotically normal with linear mean and variance and, as a consequence, the number of edges is sharply concentrated around its expected value. Moreover we prove an estimate g(q) · n −4 γ(q) n n! for the number of planar graphs with n vertices and ⌊qn⌋ edges, where γ(q) is an analytic function of q. We also show that the number of connected components in a random planar graph is distributed asymptotically as a shifted Poisson law 1 + P (ν), where ν is an explicit constant. Additional Gaussian and Poisson limit laws for random planar graphs are derived. The proofs are based on singularity analysis of generating functions and on perturbation of singularities.

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“…We will say thatM itself is series-parallel. By adapting the proof given for graphs in [5], it is not hard to see that a bipolar mapM is series-parallel if and only if it can be constructed recursively, starting from the single-edge map, by applying a sequence of series and parallel compositions:…”

Section: Permutations Avoiding 2413 and 3142 And Series-parallel Mapsmentioning

confidence: 99%

Baxter permutations and plane bipolar orientations

Bousquet-Mélou

Fusy

2008

Electronic Notes in Discrete Mathematics

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Abstract. We present a simple bijection between Baxter permutations of size n and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations (number of ascents, right-to-left maxima, left-to-right minima . . . ) into natural parameters of plane bipolar orientations (number of vertices, degree of the sink, degree of the source . . . ), and has remarkable symmetry properties. By specializing it to Baxter permutations avoiding the pattern 2413, we obtain a bijection with non-separable planar maps. A further specialization yields a bijection between permutations avoiding 2413 and 3142 and series-parallel maps.

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Enumeration and limit laws for series–parallel graphs

Cited by 70 publications

References 8 publications

Vertices of given degree in series‐parallel graphs

Vertices of given degree in series‐parallel graphs

Asymptotic enumeration and limit laws of planar graphs

Baxter permutations and plane bipolar orientations

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