Separators for Planar Graphs that are Almost Trees

TL;DR

This paper proves that connected planar graphs with slightly more edges than a tree have small vertex separators, which can be efficiently computed, aiding graph decomposition and algorithms.

Contribution

It establishes a bound on vertex separator size for planar graphs close to trees and provides a linear-time algorithm to find such separators.

Findings

Vertex separator size is O(√μ + 1) for graphs with n + μ edges.

Separator can be computed in linear time.

Applicable to planar graphs near tree structures.

Abstract

We prove that a connected planar graph with $n$ vertices and $n+\mu$ edges has a vertex separator of size $O( \sqrt{\mu} + 1)$, and this separator can be computed in linear time.