Strong Edge Geodetic Problem on Complete Multipartite Graphs and Some Extremal Graphs for the Problem

Abstract

A set of vertices X of a graph G is a strong edge geodetic set if, to any pair of vertices from X, we can assign one (or zero) shortest path between them, such that every edge of G is contained in at least one on these paths. The cardinality of a smallest strong edge geodetic set of G is the strong edge geodetic number $$\mathrm{sg_e}(G)$$ sg e ( G ) of G. In this paper, the strong edge geodetic number of complete multipartite graphs is determined. Graphs G with $$\mathrm{sg_e}(G) = n(G)$$ sg e ( G ) = n ( G ) are characterized and $$\mathrm{sg_e}$$ sg e is determined for Cartesian products $$P_n\,\square \, K_m$$ P n □ K m . The latter result in particular corrects an error from the literature.