On independent sets and non-augmentable paths in directed graphs

We investigate sufficient conditions, and in case that D be an asymmetrical digraph a necessary and sufficient condition for a digraph to have the following property: “In any induced subdigraph H of D, every maximal independent set meets every non-augmentable path”. Also we obtain a necessary and sufficient condition for any orientation of a graph G results a digraph with the above property. The property studied in this paper is an instance of the property of a conjecture of J.M. Laborde, Ch. Payan and N.H. Huang: “Every digraph contains an independent set which meets every longest directed path” (1982).