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A set of vertices $X$ of a graph $G$ is a strong edge geodetic set if to anypair of vertices from $X$ we can assign one (or zero) shortest path betweenthem 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 strongedge geodetic number ${\rm sg_e}(G)$ of $G$. In this paper, the strong edgegeodetic number of complete multipartite graphs is determined. Graphs $G$ with${\rm sg_e}(G) = n(G)$ are characterized and ${\rm sg_e}$ is determined forCartesian products $P_n\,\square\, K_m$. The latter result in particularcorrects an error from the literature.

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