
TL;DR
This paper proves that region intersection graphs over graphs excluding a complete minor have small balanced separators proportional to the square root of their edges, confirming a conjecture for string graphs and improving previous bounds.
Contribution
It establishes a universal separator size bound for region intersection graphs over minor-excluding graphs, confirming a conjecture for string graphs and extending separator theory.
Findings
String graphs with m edges have O(√m) balanced separators.
Separator size bound generalizes the planar separator theorem.
Confirms a conjecture of Fox and Pach (2010).
Abstract
For undirected graphs and , say that is a region intersection graph over if there is a family of connected subsets of such that . We show if excludes the complete graph as a minor for some , then every region intersection graph over with edges has a balanced separator with at most nodes, where is a constant depending only on . If additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. The preceding result implies that every string graph with edges has a balanced separator of size . This bound is optimal, as it generalizes the planar separator theorem. It…
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