A Separator Theorem for String Graphs and Its Applications

Authors:
Jacob Fox;János Pach
Affiliations:
Department of Mathematics, Princeton University, Princeton, USA;City College, New York,USA and EPFL, Lausanne, Switzerland
Venue:
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Year:
2009

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Abstract

A string graph is the intersection graph of a collection of continuous arcs in the plane. It is shown 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 K t ,t has at most c t n edges, where c t is a constant depending only on t . Another application is that, for any c 0, there is an integer g (c ) such that every string graph with n vertices and girth at least g (c ) has at most (1 + c )n edges.