Strong geodetic problem on Cartesian products of graphs

TL;DR

This paper investigates the strong geodetic problem on Cartesian product graphs, providing bounds, exact values for specific cases, and exploring relationships with subgraphs.

Contribution

It introduces a general upper bound for the strong geodetic number on Cartesian product graphs and computes exact values for particular graph classes.

Findings

Established an upper bound for ${ m sg}(G \,\square\, H)$

Calculated exact strong geodetic numbers for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms

Explored the relationship between strong geodetic numbers and subgraphs

Abstract

The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\rm sg}(G \,\square\, H)$ is determined, as well as exact values for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.