Strong Edge Geodetic Problem on Grids

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