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Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs

Manuel Bodirsky, Éric Fusy, Mihyun Kang, Stefan Vigerske
2007 Electronic Journal of Combinatorics  
We determine the exact and asymptotic number of unlabeled outerplanar graphs.  ...  The exact number $g_{n}$ of unlabeled outerplanar graphs on $n$ vertices can be computed in polynomial time, and $g_{n}$ is asymptotically $g\, n^{-5/2}\rho^{-n}$, where $g\approx0.00909941$ and $\rho^  ...  We now turn to the problem of asymptotic enumeration of connected outerplanar graphs.  ... 
doi:10.37236/984 fatcat:jd7sgnqbi5g7zpayq7yoboc7vi

Enumeration of Unlabeled Outerplanar Graphs [article]

Manuel Bodirsky, Stefan Vigerske
2006 arXiv   pre-print
We determine the exact and asymptotic number of unlabeled outerplanar graphs.  ...  The exact number g_n of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and g_n is asymptotically g n^-5/2ρ^-n, where g≈0.00909941 and ρ^-1≈7.50360 can be approximated.  ...  Having the asymptotic estimates of unlabeled connected outerplanar graphs and unlabeled outerplanar graphs, we investigate asymptotic distributions of parameters such as the number of components and the  ... 
arXiv:math/0511422v2 fatcat:yzrumfzlzvbjtmqarpcvqf2qpq

Asymptotic Study of Subcritical Graph Classes

Michael Drmota, Éric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Rué
2011 SIAM Journal on Discrete Mathematics  
The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number g_n/n!  ...  We present a unified general method for the asymptotic study of graphs from the so-called "subcritical" graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel  ...  Bilyana Shoilekova, Stefan Vigerske, Philippe Flajolet and Marc Noy are greatly thanked for inspiring discussions and useful comments.  ... 
doi:10.1137/100790161 fatcat:eh2dxsvv7ngdbkyvnt267lyvsi

The degree distribution in unlabelled $2$-connected graph families

Veronika Kraus
2010 Discrete Mathematics & Theoretical Computer Science  
unlabelled $2$-connected outerplanar or series-parallel graphs.  ...  International audience We study the random variable $X_n^k$, counting the number of vertices of degree $k$ in a randomly chosen $2$-connected graph of given families.  ...  Our first goal is to find a core system of functional equations with suitable properties for a subfamily of 2-connected outerplanar graphs.  ... 
doi:10.46298/dmtcs.2773 fatcat:ooff7obzlfhsvh2hgq2oxk6mdu

Boltzmann Samplers, Pólya Theory, and Cycle Pointing

Manuel Bodirsky, Éric Fusy, Mihyun Kang, Stefan Vigerske
2011 SIAM journal on computing (Print)  
The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families  ...  This is the extended and revised journal version of a conference paper with the title "An unbiased pointing operator for unlabeled structures, with applications to counting and sam-  ...  Omid Amini, Olivier Bodini, Philippe Flajolet, and Pierre Leroux are greatly thanked for fruitful discussions and suggestions. Further, we thank two anonymous referees for their helpful comments.  ... 
doi:10.1137/100790082 fatcat:yjbet7b2obditkq352p2xirphm

Boltzmann Samplers, Pólya Theory, and Cycle Pointing [article]

Manuel Bodirsky and Éric Fusy and Mihyun Kang and Stefan Vigerske
2011 arXiv   pre-print
The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families  ...  All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results.  ...  Omid Amini, Olivier Bodini, Philippe Flajolet, and Pierre Leroux are greatly thanked for fruitful discussions and suggestions. Further, we thank two anonymous referees for their helpful comments.  ... 
arXiv:1003.4546v2 fatcat:x6cznmq7ezdhdhwaajtjbubpmi

Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs

Vasiliki Velona
2018 Discrete Mathematics  
In this way, we are able to extract normal limit laws for the patterns when they are encoded, and perform asymptotic enumeration of the resulting classes when they are avoided.  ...  The results can be transfered to the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.  ...  Juanjo Rué for posing the problem and for making valuable comments on the draft. The author is also grateful to Prof. Dimitrios M. Thilikos who made this research possible.  ... 
doi:10.1016/j.disc.2018.08.027 fatcat:clqht6y4oncpzayu242ouhx3xq

Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs [article]

Vasiliki Velona
2018 arXiv   pre-print
In this way, we are able to extract normal limit laws for the patterns when they are encoded, and perform asymptotic enumeration of the resulting classes when they are avoided.  ...  The results can be directly transferred in the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.  ...  Juanjo Rué for posing the problem and for making valuable comments on the draft. The author is also grateful to Prof. Dimitrios M. Thilikos who made this research possible.  ... 
arXiv:1802.03719v2 fatcat:u6pxjxmz4zcjlg57gdxzmr65ha

Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse
2005 Journal of Graph Algorithms and Applications  
As a consequence, for rooted outerplanar maps of n nodes, we derive: • an enumeration formula, and an asymptotic of 2 3n−Θ(log n) ; • an optimal data structure of asymptotically 3n bits, built in O(n)  ...  In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges.  ...  Among graph properties, outerplanar graphs contain trees, have treewidth at most two, and are exactly the graphs of pagenumber one [4] .  ... 
doi:10.7155/jgaa.00105 fatcat:3xapxqyqs5h5bkpwbziwziizaa

Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding, and Generation [chapter]

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse
2003 Lecture Notes in Computer Science  
As a consequence, for rooted outerplanar maps of n nodes, we derive: • an enumeration formula, and an asymptotic of 2 3n−Θ(log n) ; • an optimal data structure of asymptotically 3n bits, built in O(n)  ...  In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges.  ...  Among graph properties, outerplanar graphs contain trees, have treewidth at most two, and are exactly the graphs of pagenumber one [4] .  ... 
doi:10.1007/978-3-540-39890-5_8 fatcat:srh2fz2cofe3hjfemlhkgxuyea

Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

Benedikt Stufler
2021 Electronic Journal of Combinatorics  
Our main application of these general results treats subcritical classes of unlabelled graphs. We study the special case of unlabelled outerplanar graphs in depth and calculate its scaling constant.  ...  We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions.  ...  For example, the number of labelled and unlabelled trees admit different asymptotic expressions [29] . Enumerating and studying unlabelled graphs also poses particular challenges.  ... 
doi:10.37236/9923 fatcat:4ovuurqmrfcgxlon3kh5ifbbbu

Scaling limits of random outerplanar maps with independent link-weights [article]

Benedikt Stufler
2016 arXiv   pre-print
We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us treat subclasses such as bipartite outerplanar maps.  ...  The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edge-rooted dissections of polygons.  ...  Acknowledgements I thank Nicolas Curien for discussions and feedback. I also thank an anonymous referee for helpful comments and suggestions.  ... 
arXiv:1505.07600v2 fatcat:67u7emqcljeg7dyaocxmtcbpxu

The Degree Sequence of Random Graphs from Subcritical Classes

NICLA BERNASCONI, KONSTANTINOS PANAGIOTOU, ANGELIKA STEGER
2009 Combinatorics, probability & computing  
Examples of such classes are outerplanar, series-parallel, cactus and clique graphs.  ...  Steger, 'On the degree sequences of random outerplanar and series-parallel graphs ', in APPROX-RANDOM 2008 , pp. 303-316.  ...  Acknowledgements The authors would like to express their gratitude to Oliver Riordan for his detailed comments and valuable suggestions that helped to improve the readability of our paper.  ... 
doi:10.1017/s0963548309990368 fatcat:liahgjzflbh5tmxk5t4bjxtppa

An Exact Enumeration of Distance-Hereditary Graphs [article]

Cédric Chauve and Éric Fusy and Jérémie Lumbroso
2016 arXiv   pre-print
^n---opening the perspective such graphs could be encoded on 3n bits. We also provide the exact enumeration and asymptotics of an important subclass, the 3-leaf power graphs.  ...  In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that the number of distance-hereditary  ...  Simon Fraser University; it was pursued in 2013-2014 during the third author's post-doc at the same university; the results in this paper were presented [7] at the Seventh International Conference on Graph  ... 
arXiv:1608.01464v1 fatcat:lmepfq33rvd7pd3vpjj3d2r5uy

Degree distribution in random planar graphs

Michael Drmota, Omer Gimenez, Marc Noy
2008 Discrete Mathematics & Theoretical Computer Science  
The proof uses the tools developed by Gimènez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved  ...  From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.  ...  A basic property shared by the classes of outerplanar, series-parallel and planar graphs is that a connected graph G is in the class if and only if the 2-connected components of G are also in the class  ... 
doi:10.46298/dmtcs.3562 fatcat:ik3trguefvatviqxyy2juaqlzi
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