Span of a Graph: Keeping the Safety Distance

TL;DR

This paper introduces the concept of the span of graphs to analyze the maximum safety distance two players can maintain while traversing a graph under specific move rules, providing new theoretical insights and algorithms.

Contribution

It defines novel span variants for graphs, characterizes graphs with zero safety distance, and offers a polynomial-time algorithm to compute these spans.

Findings

Defined new graph span variants related to safety distance

Characterized graphs where positive safety distance is impossible

Developed a polynomial-time algorithm for span computation

Abstract

Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, we introduce different variants of a span of a given connected graph. All the variants model the maximum safety distance kept by two players in a graph traversal, where the players may only move with accordance to a specific set of rules, and their goal: visit either all vertices, or all edges. For each variant, we show that the solution can be obtained by considering only connected subgraphs of a graph product and the projections to the factors. We characterise graphs in which it is impossible to keep a positive safety distance at all moments in time. Finally, we present a polynomial time algorithm that determines the chosen span variant of a given graph.