Iztok Banič ; Andrej Taranenko - Span of a Graph: Keeping the Safety Distance

dmtcs:9859 - Discrete Mathematics & Theoretical Computer Science, March 1, 2023, vol. 25:1 - https://doi.org/10.46298/dmtcs.9859
Span of a Graph: Keeping the Safety DistanceArticle

Authors: Iztok Banič ; Andrej Taranenko

    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.

    Volume: vol. 25:1
    Section: Graph Theory
    Published on: March 1, 2023
    Accepted on: February 19, 2023
    Submitted on: July 29, 2022
    Keywords: Mathematics - Combinatorics,05C60, 05C90

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