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

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.

25 pages, 10 figures. To appear in the Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA 2019)