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An Asymptotic Analysis of Labeled and Unlabeled k-Trees

Abstract

In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers \(U_k(n)\) of unlabeled k-trees of size n are asymptotically given by \(U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}\), where \(c_k> 0\) and \(\rho _{k}>0\) denotes the radius of convergence of the generating function \(U(z)=\sum _{n\ge 0} U_k(n) z^n\).

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Acknowledgments

We thank the anonymous reviewers for helpful suggestions on the first version of this paper.

Author information

Correspondence to Emma Yu Jin.

Additional information

The first author is partially supported by the Austrian Science Fund FWF, Project F50-02. The second author is supported by the German Research Foundation DFG, Project JI 207/1-1 and Austrian Research Fund FWF, SFB F50 Algorithmic and Enumerative Combinatorics.

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Drmota, M., Jin, E.Y. An Asymptotic Analysis of Labeled and Unlabeled k-Trees. Algorithmica 75, 579–605 (2016). https://doi.org/10.1007/s00453-015-0039-1

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Keywords

  • k-trees
  • Generating function
  • Singularity analysis
  • Central limit theorem

Mathematics Subject Classification

  • 05A16
  • 05A15