 
Summary: Independent transversals of longest paths in locally
semicomplete and locally transitive digraphs
Hortensia GaleanaS´anchez, Ricardo G´omez
and Juan Jos´e MontellanoBallesteros
Instituto de Matem´aticas de la Universidad Nacional Aut´onoma de M´exico
Circuito Exterior, Ciudad Universitaria C.P. 04510, M´exico D.F., M´exico
Abstract
We present several results concerning the LabordePayanXuang
conjecture stating that in every digraph there exists an independent set
of vertices intersecting every longest path. The digraphs we consider
are defined in terms of local semicompleteness and local transitivity.
We also look at oriented graphs for which the length of a longest path
does not exceed 4.
Keywords: Independent set; longest path; locally semicomplete; lo
cally transitive
2000 Mathematics Subject Classification: 05C20
1 Introduction
Given a digraph, does there exist a maximal independent set of vertices
that is transversal to all longest paths (every longest path has a vertex on
the set)? This question was posed by Laborde, Payan and Xuong in 1982
