Strong edge geodetic problem on grids

TL;DR

This paper investigates the strong edge geodetic problem on grid graphs, providing exact values for certain cases and establishing upper bounds for the strong edge geodetic number on Cartesian products of paths.

Contribution

It computes the exact strong edge geodetic number for specific grid graphs and offers general upper bounds for these numbers on Cartesian products of paths.

Findings

Exact values of sge for P_n × P_2, P_n × P_3, and P_n × P_4.

General upper bounds for sge on P_n × P_m.

Insights into the structure of strong edge geodetic sets on grid graphs.

Abstract

Let $G=(V(G),E(G))$ be a simple graph. A set $S \subseteq V(G)$ is a strong edge geodetic set if there exists an assignment of exactly one shortest path between each pair of vertices from $S$, such that these shortest paths cover all the edges $E(G)$. The cardinality of a smallest strong edge geodetic set is the strong edge geodetic number $\text{sge}(G)$ of $G$. In this paper, the strong edge geodetic problem is studied on the Cartesian product of two paths. The exact value of the strong edge geodetic number is computed for $P_n \, \square \, P_2$, $P_n \, \square \, P_3$ and $P_n \, \square \, P_4$. Some general upper bounds for $\text{sge}(P_n \, \square \, P_m)$ are also proved.