Planar Digraphs without Large Acyclic Sets

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 ng3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most remvoe<n(g-2)+1g-1. When g=3 this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible. (C) 2016 Wiley Periodicals, Inc.