Planar digraphs without large acyclic sets
Kolja Knauer, Petru Valicov, Paul S. Wenger

TL;DR
This paper constructs specific planar directed graphs demonstrating that the largest acyclic set can be at most roughly half the vertices, disproving a previous conjecture and answering an open question.
Contribution
It provides explicit examples of planar digraphs with small maximum acyclic sets, disproving a conjecture and confirming the optimality of a known bound.
Findings
Existence of planar digraphs with maximum acyclic set size at most (n+1)/2
Disproof of Harutyunyan's conjecture
Confirmation that Albertson's question is best possible
Abstract
Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order for which the size of the maximum acyclic set is at most , for any . This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
