Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size

DROPS (Schloss Dagstuhl – Leibniz Center for Informatics), pp. 13

Abstract

Preferential attachment graphs are random graphs designed to mimic properties of real word networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We prove various structural asymptotic properties of this graph model. In particular, we show that the size of the largest r-shallow clique minor in Gⁿ_m is at most log(n)^{O(r²)}m^{O(r)}. Furthermore, there exists a one-subdivided clique of size log(n)^{1/4}. Therefore, preferential attachment graphs are asymptotically almost surely somewhere dense and algorithmic techniques developed for structurally sparse graph classes are not directly applicable. However, they are just barely somewhere dense. The removal of just slightly more than a polylogarithmic number of vertices asymptotically almost surely yields a graph with locally bounded treewidth.

Authors 3

1. Jan Dreier Aachen Department of Computer Science

   RWTH Aachen University

   Affiliation as printed

   Department of Computer Science, RWTH Aachen University, Germany

   RWTH Aachen University, Aachen, Germany

2. Philipp Kuinke Aachen Department of Computer Science

   RWTH Aachen University

   Affiliation as printed

   Department of Computer Science, RWTH Aachen University, Germany

   RWTH Aachen University, Aachen, Germany

3. Peter Rossmanith Aachen Department of Computer Science

   RWTH Aachen University

   Affiliation as printed

   Department of Computer Science, RWTH Aachen University, Germany

   RWTH Aachen University, Aachen, Germany