Extremal k-forcing sets in oriented graphs

This article studies the k-forcing number for oriented graphs, generalizing both the zero forcing number for directed graphs and the k-forcing number for simple graphs. In particular, given a simple graph G, we introduce the maximum (minimum) oriented k-forcing number, denoted MOFk(G) (mofk(G)), which is the largest (smallest) k-forcing number among all possible orientations of G. These new ideas are compared to known graph invariants and it is shown that, among other things, mof(G) equals the path covering number of G while MOFk(G) is greater than or equal to the independence number of G — with equality holding if G is a tree or if k is at least the maximum degree of G. Along the way, we also show that many recent results about k-forcing number can be modified for oriented graphs.