ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 33(2) (2013) 261-275
DOI: 10.7151/dmgt.1650

Independent Detour Transversals in 3-deficient Digraphs

Susan van Aardt, Marietjie Frick
Joy Singleton

Department of Mathematical Sciences
University of South Africa
P.O. Box 392, Unisa, 0003, South Africa


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.


[1]S.A. van Aardt, G. Dlamini, J. Dunbar, M. Frick, and O. Oellermann, The directed path partition conjecture, Discuss. Math. Graph Theory 25 (2005) 331--343, doi: 10.7151/dmgt.1286.
[2]S.A. van Aardt, J.E. Dunbar, M. Frick, P. Katrenič, M.H. Nielsen, and O.R. Oellermann, Traceability of k-traceable oriented graphs, Discrete Math. 310 (2010) 1325--1333, doi: 10.1016/j.disc.2009.12.022.
[3]J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer-Verlag, London, 2001).
[4]J. Bang-Jensen, M.H. Nielsen and A. Yeo, Longest path partitions in generalizations of tournaments, Discrete Math. 306 (2006) 1830--1839, doi: 10.1016/j.disc.2006.03.063.
[5]J.A. Bondy, Basic graph theory: Paths and circuits, in: Handbook of Combinatorics, ed(s), R.L. Graham, M. Grötschel and L. Lovász The MIT Press, Cambridge, MA, 1995) Vol I, p. 20.
[6]P. Camion, Chemins et circuits hamiltoniens des graphes complets, C.R. Acad. Sci. Paris 249 (1959) 2151--2152.
[7]C.C. Chen and P. Manalastas Jr., Every finite strongly connected digraph of stability 2 has a Hamiltonian path, Discrete Math. 44 (1983) 243--250, doi: 10.1016/0012-365X(83)90188-7.
[8]H. Galeana-Sánchez and R. Gómez, Independent sets and non-augmentable paths in generalizations of tournaments, Discrete Math. 308 (2008) 2460--2472, doi: 10.1016/j.disc.2007.05.016.
[9]H. Galeana-Sánchez and H.A. Rincón-Mejía, Independent sets which meet all longest paths, Discrete Math. 152 (1996) 141--145, doi: 10.1016/0012-365X(94)00261-G.
[10]F. Havet, Stable set meeting every longest path, Discrete Math. 289 (2004) 169--173, doi: 10.1016/j.disc.2004.07.013.
[11]J.M. Laborde, C. Payan, N.H. Xuong, Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173--177 (Teubner-Texte Math., 59 1983).
[12]M. Richardson, Solutions of irreflexive relations, Ann. of Math. 58 (1953) 573--590, doi: 10.2307/1969755.

Received 19 October 2011
Revised 28 February 2012
Accepted 28 February 2012