Local Structure Theorems for Erdős–Rényi Graphs and Their Algorithmic Applications

  • Jan Dreier
  • Philipp Kuinke
  • Ba Le Xuan
  • Peter Rossmanith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10706)

Abstract

We analyze local properties of sparse Erdős–Rényi graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths. For \(d(n)=n^{o(1)}\) we show that \(G(n,d(n)/n)\) has asymptotically almost surely (a.a.s.) bounded local treewidth and therefore is a.a.s. nowhere dense. We also discover a new and simpler proof that \(G(n,d/n)\) has a.a.s. bounded expansion for constant d. The local structure of sparse Erdős–Rényi graphs is very special: The r-neighborhood of a vertex is a tree with some additional edges, where the probability that there are m additional edges decreases with m. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally, experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.

Keywords

Graph theory Random graphs Sparse graphs Graph algorithms 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Jan Dreier
    • 1
  • Philipp Kuinke
    • 1
  • Ba Le Xuan
    • 2
  • Peter Rossmanith
    • 1
  1. 1.Theoretical Computer Science, Department of Computer ScienceRWTH Aachen UniversityAachenGermany
  2. 2.The Sirindhorn International Thai-German Graduate School of EngineeringKing Mongkut’s University of Technology North BangkokBangkokThailand

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