Some considerations on the maximal safety distance in a graph

TL;DR

This paper explores the concept of maximal safety distance in graphs, providing a new approach to measure and analyze safety distances for moving actors, and offers solutions for specific graph classes.

Contribution

It introduces a novel method for defining and calculating the maximal safety distance in graphs, differing from previous span-based approaches.

Findings

Solved several safety distance problems in graphs

Established relations between different graph spans

Calculated span values for specific graph classes

Abstract

The work in this paper is motivated by I. Bani\v{c}'s and A. Taranenko's recent paper, where they introduced a new notion, the span of a graph. Their goal was to solve the problem of keeping a safety distance while two actors are moving through a graph and they present three different types of graph spans, depending on the movement rules. We observe the same goal, but give a different approach to that problem by directly defining the maximal safety distance for different movement rules two actors can take. This allowed us to solve several problems, prove some relations between different graph spans and calculate the span values for some classes of graphs.