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

A set image of vertices of a graph image is a geodetic set if every vertex of image lies in at least one interval between the vertices of image. The size of a minimum geodetic set in image is the geodetic number of image. 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.