Convexity in directed graphs

In this paper the concept of convexity in directed graphs is described. It is shown that the set of convex subgraphs of a directed graph G partially ordered by inclusion forms a complete, semimodular, A-regular lattice, denoted ?_G. The lattice theoretic properties of the convex subgraph lattice lead to inferences about the path structure of the original graph G. In particular, a graph factorization theorem is developed. In Section 4, several graph homomorphism concepts are investigated in relation to the preservation of convexity properties. Finally we characterize an interesting class of locally convex directed graphs.