Minimum diameter and cycle-diameter orientations on planar graphs

TL;DR

This paper investigates orientations of planar graphs that minimize the cycle diameter, establishing bounds and implications for approximation algorithms in graph theory.

Contribution

It proves that planar graphs always admit orientations with cycle diameters bounded by a constant times the original, advancing understanding of graph orientation properties.

Findings

Cycle diameter bounds are established for planar graphs.

Orientation with bounded cycle diameter exists for planar graphs.

Implications for approximation algorithms in graph orientation problems.

Abstract

Let G be an edge weighted undirected graph. For every pair of nodes consider the shortest cycle containing these nodes in G. The cycle diameter of G is the maximum length of a cycle in this set. Let H be a directed graph obtained by directing the edges of G. The cycle diameter of H is similarly defined except for that cycles are replaced by directed closed walks. Is there always an orientation H of G whose cycle diameter is bounded by a constant times the cycle diameter of G? We prove this property for planar graphs. These results have implications on the problem of approximating an orientation with minimum diameter