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.

Comment: Discrete Mathematics and Theoretical Computer Science vol. 25:1 #8 (2023)

• 05C12 - Distance in graphs
• 05C57 - Games on graphs (graph-theoretic aspects)
• 05C76 - Graph operations (line graphs, products, etc.)
• 91A43 - Games involving graphs