Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs

TL;DR

This paper presents an efficient algorithm for finding the shortest set of vertex-disjoint paths connecting specified vertices incident to two faces in a directed planar graph, optimizing total path length.

Contribution

It introduces a novel $O(kn ext{log} n)$ time algorithm for computing minimal total length vertex-disjoint paths between face-incident vertices in planar graphs.

Findings

Algorithm runs in $O(kn ext{log} n)$ time.

Successfully computes shortest vertex-disjoint paths in planar graphs.

Applicable to network routing and graph optimization problems.

Abstract

Let $G$ be a directed planar graph of complexity $n$, each arc having a nonnegative length. Let $s$ and $t$ be two distinct faces of $G$; let $s_1,...,s_k$ be vertices incident with $s$; let $t_1,...,t_k$ be vertices incident with $t$. We give an algorithm to compute $k$ pairwise vertex-disjoint paths connecting the pairs $(s_i,t_i)$ in $G$, with minimal total length, in $O(kn\log n)$ time.