On the geodetic number and related metric sets in Cartesian product graphs

A set SS of vertices of a graph GG is a geodetic set if every vertex of GG lies in at least one interval between the vertices of SS. The size of a minimum geodetic set in GG is the geodetic number of GG. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products have product structure and that the contour set of a Cartesian product is geodetic if and only if their projections are geodetic sets in factors.