An efficient algorithm for packing cuts and (2,3)-metrics in a planar graph with three holes

TL;DR

This paper presents a strongly polynomial combinatorial algorithm for packing cuts and (2,3)-metrics in a planar graph with three holes, ensuring the realization of distances within each hole.

Contribution

It introduces a new efficient algorithm for packing cuts and (2,3)-metrics in planar graphs with three holes, improving computational methods in this area.

Findings

Algorithm is strongly polynomial and purely combinatorial.

Successfully realizes distances within each hole.

Applicable to planar graphs with specific cycle conditions.

Abstract

We consider a planar graph $G$ in which the edges have nonnegative integer lengths such that the length of every cycle of $G$ is even, and three faces are distinguished, called holes in $G$. It is known that there exists a packing of cuts and (2,3)-metrics with nonnegative integer weights in $G$ which realizes the distances within each hole. We develop a strongly polynomial purely combinatorial algorithm to find such a packing.