On the geodetic iteration number of a graph in which geodesic and monophonic convexities are equivalent
Let G be a graph, u and v two vertices of G, and X a subset of V(G). A u-v geodesic is a path between u and v of minimum length. Ig(u,v) is the set of vertices that lie on any u-v geodesic and Ig(X) is the set ⋃u,v∈XIg(u,v). X is g-convex if Ig(X)=X. Analogously, Im(u,v) is the set of vertices that lie on any induced path between u and v and Im(X) is the set ⋃u,v∈XIm(u,v). X is m-convex if Im(X)=X. The g-convex hull [X]g of X is the smallest g-convex set containing X. Igh(X) equals Ig(X), if h=1, and equals I(Igh−1(X)), if h>1.