On the Degree Sequences of Random Outerplanar and Series-Parallel Graphs

Bernasconi, Nicla; Παναγιώτου, Κωνσταντίνος; Steger, Angelika

doi:10.1007/978-3-540-85363-3_25

Cited by 16 publications

(71 citation statements)

References 11 publications

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“…n be the number of edges in the SP graph of color r after the random insertion of n edges, and let X n be the vector with the s + 1 components X (1) n , X (2) n , . .…”

Section: Nodes Of Small Outdegreementioning

confidence: 99%

“…For color 1, we write the conditional recurrence (2) n | F n−1 ] . Let F n be the sigma field generated by the the first n edge insertions.…”

Section: Nodes Of Small Outdegreementioning

confidence: 99%

“…To the best of our knowledge, these networks have been studied under two models of randomness: the uniform model, where all SP networks of a certain size are equally likely [2,4], and the hierarchical lattice model [5]. To the best of our knowledge, these networks have been studied under two models of randomness: the uniform model, where all SP networks of a certain size are equally likely [2,4], and the hierarchical lattice model [5].…”

Section: Introductionmentioning

confidence: 99%

“…Therefor it suffices to get the results for X (1) n and X (2) n . Therefor it suffices to get the results for X (1) n and X (2) n .…”

mentioning

confidence: 99%

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Some Node Degree Properties of Series–parallel Graphs Evolving Under a Stochastic Growth Model

Mahmoud¹

2013

Prob. Eng. Inf. Sci.

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We introduce a natural growth model for directed series-parallel (SP) graphs and look at some of the graph properties under this stochastic model. Specifically, we look at the degrees of certain types of nodes in the random SP graph. We examine the degree of a pole and will find its exact distribution, given by a probability formula with alternating signs. We also prove that, for a fixed value s, the number of nodes of outdegree 1, . . . , s asymptotically has a joint multivariate normal distribution. Pólya urns will systematically provide a working tool.

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“…n be the number of edges in the SP graph of color r after the random insertion of n edges, and let X n be the vector with the s + 1 components X (1) n , X (2) n , . .…”

Section: Nodes Of Small Outdegreementioning

confidence: 99%

“…For color 1, we write the conditional recurrence (2) n | F n−1 ] . Let F n be the sigma field generated by the the first n edge insertions.…”

Section: Nodes Of Small Outdegreementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Therefor it suffices to get the results for X (1) n and X (2) n . Therefor it suffices to get the results for X (1) n and X (2) n .…”

mentioning

confidence: 99%

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Some Node Degree Properties of Series–parallel Graphs Evolving Under a Stochastic Growth Model

Mahmoud¹

2013

Prob. Eng. Inf. Sci.

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“…Using the concept of Boltzmann samplers, cf. Section 2.2, Bernasconi, Panagiotou, and Steger [4,5] obtained for certain subclasses of planar graphs not only the maximum degree, but also the degree distribution of an element drawn uniformly at random from this class. Their approach relied implicitly on the fact that the classes under consideration have a simple block structure, i.e.…”

Section: Introductionmentioning

confidence: 99%

Maximal Biconnected Subgraphs of Random Planar Graphs

Παναγιώτου¹,

Steger²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Let Pn be the class of simple labeled planar graphs with n vertices, and denote by Pn a graph drawn uniformly at random from this set. Basic properties of Pn were first investigated by Denise, Vasconcellos, and Welsh [7]. Since then, the random planar graph has attracted considerable attention, and is nowadays an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we study closely the structure of Pn. More precisely, let b( ; Pn) be the number of blocks (i.e. maximal biconnected subgraphs) of Pn that contain exactly vertices, and let lb(Pn) be the number of vertices in the largest block of Pn. We show that with high probability Pn contains a giant block that includes up to lower order terms cn vertices, where c ≈ 0.959 is an analytically given constant. Moreover, we show that the second largest block contains only

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Vertices of degree k in random unlabeled trees

Παναγιώτου

Sinha

2011

Journal of Graph Theory

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Let H n be the class of unlabeled trees with n vertices, and denote by H n a tree that is drawn uniformly at random from this set. The asymptotic behavior of the random variable deg k (H n ) that counts vertices of degree k in H n was studied, among others, by Drmota and Gittenberger in [J Graph Theory 31(3) (1999), , who showed that this quantity satisfies a central limit theorem. This result provides a very precise characterization of the "central region" of the distribution, but does not give any non-trivial information about its tails. In this work, we study further the number of vertices of degree k in H n . In particular, for k = O((log n / (log log n)) 1/2 ) we show exponential-type bounds for the probability that deg k (H n ) deviates from its expectation. On the technical side, our proofs are based on the analysis of a randomized algorithm that generates unlabeled trees in the so-called Boltzmann model. The analysis of such

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On the Degree Sequences of Random Outerplanar and Series-Parallel Graphs

Cited by 16 publications

References 11 publications

Some Node Degree Properties of Series–parallel Graphs Evolving Under a Stochastic Growth Model

Some Node Degree Properties of Series–parallel Graphs Evolving Under a Stochastic Growth Model

Maximal Biconnected Subgraphs of Random Planar Graphs

Vertices of degree k in random unlabeled trees

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