### Products of Geodesic Graphs and the Geodetic Number of Products

Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(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 Km,n with itself, where m, n ≥ 4, is equal to the minimum of m, n and eight.