# Planar digraphs without large acyclic sets

Abstract : Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers $n\geq g\geq 3$, there exist oriented planar graphs of order $n$ and digirth $g$ for which the size of the maximum acyclic set is at most $\lceil \frac{n(g-2)+1}{g-1} \rceil$. When $g=3$ this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-01198779
Contributor : Petru Valicov <>
Submitted on : Monday, September 14, 2015 - 1:33:19 PM
Last modification on : Monday, March 4, 2019 - 2:04:14 PM

### Citation

Kolja Knauer, Petru Valicov, Paul Wenger. Planar digraphs without large acyclic sets. Journal of Graph Theory, Wiley, 2016, ⟨10.1002/jgt.22061⟩. ⟨hal-01198779⟩

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