Strong edge geodetic problem on grids

Abstract

Let G = (V (G), E(G)) be a simple graph. A set S ⊆ 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 sg e (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 P 2 , P n P 3 and P n P 4 . Some general upper bounds for sg e (P n P m ) are also proved.