Algorithms for generating convex sets in acyclic digraphs

A set X of vertices of an acyclic digraph D is convex if X ̸= ∅ and there is no directed path between vertices of X which contains a vertex not in X. A set X is connected if X ̸= ∅ and the underlying undirected graph of the subgraph ofD induced byX is connected. Connected convex sets and convex sets of acyclic digraphs are of interest in the area of modern embedded processor technology. We construct an algorithm A for enumeration of all connected convex sets of an acyclic digraph D of order n. The time complexity of A is O(n · cc(D)), where cc(D) is the number of connected convex sets in D. We also give an optimal algorithm for enumeration of all (not just connected) convex sets ∗Department of Mathematical Sciences, University of Memphis, TN 38152-3240, USA, E-mail: [email protected] †Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected] ‡Department of Computer Science, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected] §Department of Computer Science, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected] ¶Department of Computer Science, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected] ∥Department of Computer Science, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected] ∗∗Department of Computer Science, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected] ††Department of Computer Science, Royal Holloway, University of London, Egham, TW20 0EX, UK, E-mail: [email protected]