A Separator Theorem for String Graphs and its Applications

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(m3/4 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 > 0, there is an integer g( ) such that every string graph with n vertices and girth at least g( ) has at most (1 + )n edges.

• Type: Journal
• Year: 2010
• Where: CPC
• Authors: Jacob Fox, János Pach
• Keywords: Complete Bipartite Subgraph, CPC 2010, Operations Research, String Graph, Tree-like String Graphs