On the number of connected convex subgraphs of a connected acyclic digraph

A digraph image is connected if the underlying undirected graph of image is connected. A subgraph image of an acyclic digraph image is convex if there is no directed path between vertices of image which contains an arc not in image. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order image. Connected convex subgraphs of connected acyclic digraphs are of interest in the area of modern embedded processors technology.