Representation of short distances in structurally sparse graphs

TL;DR

This paper introduces a new way to efficiently represent short paths in sparse graphs using partial orientations called weak guidance systems, applicable to many natural graph classes, with algorithmic insights.

Contribution

It defines weak guidance systems for sparse graphs, proves their existence in various classes, and explores their algorithmic properties.

Findings

Many natural graph classes admit weak guidance systems.

Bounded outdegree guidance systems efficiently represent short paths.

Algorithmic methods for constructing guidance systems are studied.

Abstract

A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.