Planar Digraphs without Large Acyclic Sets
نویسندگان
چکیده
منابع مشابه
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 n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most dn(g−2)+1 g−1 e. When g = 3 this result disproves a conjecture of Harutyunyan and shows that a question of Albert...
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An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2-cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if g is the length of the shortest directed cycle, we show that...
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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 CO(D) (CC(D)) and its size by co(D) (cc(D)). Gutin, Johnstone...
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Let G be a directed planar graph on n vertices, with no directed cycle of length less than g > 4. We prove that G contains a set X of vertices such that G−X has no directed cycle, and |X| 6 5n−5 9 if g = 4, |X| 6 2n−5 4 if g = 5, and |X| 6 2n−6 g if g > 6. This improves recent results of Golowich and Rolnick. A directed graph G (or digraph, in short) is said to be acyclic if it does not contain...
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ژورنال
عنوان ژورنال: Journal of Graph Theory
سال: 2016
ISSN: 0364-9024
DOI: 10.1002/jgt.22061