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A Separator Theorem for String Graphs and its Applications

Published online by Cambridge University Press:  07 October 2009

JACOB FOX  and
JÁNOS PACH
JACOB FOX
Affiliation:
Department of Mathematics, Princeton, NJ 08544, USA (e-mail: jacobfox@math.princeton.edu)
JÁNOS PACH
Affiliation:
EPFL Lausanne, IMB-DCG, Station 8, CH-1015, Switzerland and Rényi Institute, Budapest, H-1364, Hungary (e-mail: pach@cims.nyu.edu)
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Abstract

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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.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 3 , May 2010 , pp. 371 - 390
Copyright
Copyright © Cambridge University Press 2009

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