Span of a Graph: Keeping the Safety Distance

Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce novel notion graphs. Using this, solve problem determining \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, different variants given connected All model maximum distance kept in graph traversal, where may only accordance specific set rules, goal: visit either vertices, or edges. each variant, show that solution obtained considering subgraphs product projections factors. We characterise graphs which it is impossible positive moments time. Finally, present polynomial time algorithm determines chosen variant