Abstract
A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by \(\mathcal{CO}(D) \ (\mathcal{CC}(D))\) and its size by co(D) (cc(D)). Gutin et al. (2008) conjectured that the sum of the sizes of all convex sets (connected convex sets) in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with \(\sum_{C\in \mathcal{CO}(D)}|C| = o(n\cdot \mathrm{co}(D))\) and \(\sum_{C\in \mathcal{CC}(D)}|C| = o(n\cdot \mathrm{cc}(D))\). We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n − k + 1. This is a strengthening of a theorem of Gutin and Yeo.
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Balister, P., Gerke, S. & Gutin, G. Convex Sets in Acyclic Digraphs. Order 26, 95–100 (2009). https://doi.org/10.1007/s11083-009-9109-9
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DOI: https://doi.org/10.1007/s11083-009-9109-9