Finding Maximal Sets of Laminar 3-Separators in Planar Graphs in Linear Time

TL;DR

This paper presents a linear-time algorithm for finding maximal laminar sets of 3-separators in 3-connected planar graphs, aiding in graph decomposition and disjoint path algorithms.

Contribution

It introduces a novel linear-time method for identifying maximal laminar 3-separators and constructing related tree decompositions in planar graphs.

Findings

Linear-time algorithm for laminar 3-separators

Construction of tree decompositions of adhesion three

Applications in disjoint path algorithms

Abstract

We consider decomposing a 3-connected planar graph $G$ using laminar separators of size three. We show how to find a maximal set of laminar 3-separators in such a graph in linear time. We also discuss how to find maximal laminar set of 3-separators from special families. For example we discuss non-trivial cuts, ie. cuts which split $G$ into two components of size at least two. For any vertex $v$, we also show how to find a maximal set of 3-separators disjoint from $v$ which are laminar and satisfy: every vertex in a separator $X$ has two neighbours not in the unique component of $G-X$ containing $v$. In all cases, we show how to construct a corresponding tree decomposition of adhesion three. Our new algorithms form an important component of recent methods for finding disjoint paths in nonplanar graphs.