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Independent sets and non-augmentable paths in generalizations of tournaments

Summary: Independent sets and non-augmentable paths
in generalizations of tournaments
Hortensia Galeana-S´anchez and Ricardo G´omez
Instituto de Matem´aticas de la Universidad Nacional Aut´onoma de M´exico.
´Area de la Investigaci´on Cient´ifica. Circuito Exterior, Ciudad Universitaria.
M´exico D.F. 04510 M´EXICO
We study different classes of digraphs, which are generalizations of tournaments,
to have the property of possessing a maximal independent set intersecting every
non-augmentable path (in particular, every longest path). The classes are the arc-
local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and
semicomplete k-partite digraphs. We present results on strongly internally and fi-
nally non-augmentable paths as well as a result that relates the degree of vertices
and the length of longest paths. A short survey is included in the introduction.
Key words: Independent set; non-augmentable path; longest path; generalization
of tournament
1 Introduction
The conjecture of Laborde, Payan and Xuong can be stated as follows: In every
digraph, there exists a maximal independent set that intersects every longest
path (see [21]). The conjecture is true for every digraph having a kernel, that


Source: Aíza, Ricardo Gómez - Instituto de Matemáticas, Universidad Nacional Autónoma de México


Collections: Mathematics