Geodesic
Independent Detour Transversals in 3-Deficient Digraphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 261-275

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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 
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van Aardt, Susan; Frick, Marietjie; Singleton, Joy. Independent Detour Transversals in 3-Deficient Digraphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 261-275. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a1/