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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 , there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most . When this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best...
In this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound of Alon–Kahn–Seymour for , and implies that such a graph has an...
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