Characteristic properties and recognition of graphs in which geodesic and monophonic convexities are equivalent

Let G be a connected graph. A subset X of V(G) is g-convex (mconvex) if it contains all vertices on shortest (induced) paths between vertices in X. We state characteristic properties of graphs in which every g-convex set is m-convex, based on which we show that such graphs can be recognized in polynomial time. Moreover, we state a new convexity - theoretic characterization of Ptolemaic graphs