Products of Geodesic Graphs and the Geodetic Number of Products

Given a connected graph and a vertex $x ∈ V (G)$, the geodesic graph $P_x(G)$ has the same vertex set as $G$ with edges $uv$ iff either $v$ is on an $x − u$ geodesic path or $u$ is on an $x − v$ geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of $K_{m,n}$ with itself, where $m, n ≥ 4$, is equal to the minimum of $m, n$ and eight.