Strong geodetic problem on Cartesian products of graphs

Abstract

The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number $\mathrm{sg}(G)$ 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 $\mathrm{sg}(G\square H)$ 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.

18 pages, 9 figures