A separator theorem for string graphs and its applications

  • Authors:

  • Jacob Fox;JÁnos Pach

  • Affiliations:

  • Department of mathematics, princeton, nj 08544, usa (e-mail: jacobfox@math.princeton.edu);Epfl lausanne, imb-dcg, station 8, ch-1015, switzerland and rényi institute, budapest, h-1364, hungary (e-mail: pach@cims.nyu.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2010

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Abstract

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ε 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 + ε)n edges. Furthermore, the number of such labelled graphs is at most (1 + ε)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.