Non-Crossing Shortest Paths are Covered with Exactly Four Forests

TL;DR

This paper proves that a set of non-crossing shortest paths in a planar graph can be covered with exactly four forests, providing a tight bound for the minimal number of forests needed.

Contribution

It establishes a tight upper bound of four forests for covering non-crossing shortest paths in certain planar graphs, advancing understanding of path covering in graph theory.

Findings

PCFN(P)  4 for non-crossing shortest paths in planar graphs

The bound of four forests is tight

Applicable when extremal vertices lie on the same face

Abstract

Given a set of paths $P$ we define the \emph{Path Covering with Forest Number} of $P$} (PCFN($P$)) as the minimum size of a set $F$ of forests satisfying that every path in $P$ is contained in at least one forest in $F$. We show that PCFN($P$) is treatable when $P$ is a set of non-crossing shortest paths in a plane graph or subclasses. We prove that if $P$ is a set of non-crossing shortest paths of a planar graph $G$ whose extremal vertices lie on the same face of $G$, then PCFN($P$)\leq 4$, and this bound is tight.