Generalization of edge general position problem

The edge geodesic cover problem of a graph G is to find a smallest number of geodesics that cover the edge set of G. The edge k-general position problem is introduced as the problem to find a largest set S of edges of G such that at most k-1 edges of S lie on a common geodesic. We show that these are dual min-max problems and connect them to an edge geodesic partition problem. Using these connections, exact values of the edge k-general position number is determined for different values of k and for various networks including torus networks, hypercubes, and Benes networks.