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Discussiones Mathematicae Graph Theory 33(2) (2013)
261-275
DOI: https://doi.org/10.7151/dmgt.1650
Independent Detour Transversals in 3-deficient Digraphs
Susan van Aardt, Marietjie Frick and Joy Singleton
Department of Mathematical Sciences |
Abstract
In 1982 Laborde, Payan and Xuong [Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173-177 (Teubner-Texte Math., 59 1983)] conjectured that every digraph has an independent detour transversal (IDT), i.e. an independent set which intersects every longest path. Havet [Stable set meeting every longest path, Discrete Math. 289 (2004) 169-173] showed that the conjecture holds for digraphs with independence number two. A digraph is p-deficient if its order is exactly p more than the order of its longest paths. It follows easily from Havet's result that for p = 1,2 every p-deficient digraph has an independent detour transversal. This paper explores the existence of independent detour transversals in 3-deficient digraphs.
Keywords: longest path, independent set, detour transversal, strong digraph, oriented graph
2010 Mathematics Subject Classification: 05C20, 05C38.
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Received 19 October 2011
Revised 28 February 2012
Accepted 28 February 2012
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