The relation between different edge spans of a graph
Aljoša Šubašić, Tanja Vojković
TL;DR
This work studies edge-span variants in finite connected graphs, defining strong, direct, and Cartesian edge spans $\sigma^{\boxtimes}_{E}(G)$, $\sigma^{\times}_{E}(G)$, and $\sigma^{\square}_{E}(G)$ as edge-visit analogues to vertex spans. It develops a framework using $l$-sweeps and opposite lazy tracks to bound and relate these spans, proving that $\max\{\sigma^{\times}_{E}(G),\sigma^{\square}_{E}(G)\} \le \sigma^{\boxtimes}_{E}(G) \le \max\{\sigma^{\times}_{E}(G),\sigma^{\square}_{E}(G)\}+1$, with tightness shown by the graph family $K_n^+$ where $\sigma^{\boxtimes}_{E}(K_n^+)=2$ and the other two spans equal $1$ (for $n\ge4$). The authors also determine exact spans for these graphs and establish that, unlike in some vertex-span cases, the strong edge span can exceed the maximum of the other two, thereby refining the theory of graph safety-distance variants. They further relate edge-span results to vertex-span behavior and conjecture a similar equality for vertex spans, highlighting open problems and future directions. Overall, the paper extends vertex-span results to edge spans, clarifies when the strong variant tightens bounds, and provides concrete constructions and exact values essential for theoretical and potential applied analyses of graph-based safety-distance problems.
Abstract
In several recent papers, the maximal safety distance that two players can maintain while moving through a graph has been defined and studied using three different spans of the graph, each with different movement conditions. Mainly, vertex spans have been studied, in which players visit all the vertices of a graph. In this paper, we analyze the values of three edge spans, which represent the maximal safety distance that two players can maintain while visiting all the edges of a graph. We present edge span values for some graph classes and examine the relationship between different variants of edge spans.