Geodetic metrizations of graphs

A graph can be metrized by assigning a length to each of its edges. Such a graph is said to be geodetic if for each pair of vertices there is a unique geodesic joining them. It is said to be normally geodetic if each of these unique geodesics is one of the geodesics in the usual metrization of the graph in which each edge is given unit length. It is shown that every graph admits a normally geodetic metrization. Geodetic metrizations of the four- and eight-connection graphs of the digital plane which can be processed easily on a computer are investigated. Examples are given of normally geodetic integral metrizations of arbitrarily large rectangular blocks of these planes. However, it is proved that there are no normally geodetic metrizations of the whole of these planes which are periodic in each variable.