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The Giant Component and 2-Core in Sparse Random Outerplanar Graphs

Authors Mihyun Kang, Michael Missethan

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Generating functions
Keywords
  • giant component
  • core
  • outerplanar graphs
  • singularity analysis

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Abstract

Let A(n,m) be a graph chosen uniformly at random from the class of all vertex-labelled outerplanar graphs with n vertices and m edges. We consider A(n,m) in the sparse regime when m=n/2+s for s=o(n). We show that with high probability the giant component in A(n,m) emerges at m=n/2+O (n^{2/3}) and determine the typical order of the 2-core. In addition, we prove that if s=ω(n^{2/3}), with high probability every edge in A(n,m) belongs to at most one cycle.

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Mihyun Kang and Michael Missethan. The Giant Component and 2-Core in Sparse Random Outerplanar Graphs. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.AofA.2020.18

Author Details

Mihyun Kang
  • Institute of Discrete Mathematics, Graz University of Technology, Austria
Michael Missethan
  • Institute of Discrete Mathematics, Graz University of Technology, Austria

Funding

  • Kang, Mihyun: Research supported by the Austrian Science Fund (FWF): I3747, W1230.
  • Missethan, Michael: Research supported by the Austrian Science Fund (FWF): W1230.

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