Independent sets which meet all longest paths

We prove some sufficient conditions for a directed graph to have the property of a conjecture of J. (1982): "Every directed graph contains an independent set which meets every longest directed path". Let G be a directed graph, and denote by V(G) its vertex-set, by A(G) its arc-set, X(G) denotes its chromatic number, and A(G) the length of the longest directed path. Independently, B. Roy and T. Gallai proved that X (G) :::::; A(G). Consider an independent set S ('stable' set), and denote by G-S the subgraph of G induced by V(G)-S; in 1982, Laborde, Payan and Huang conjectured a plausible looking extension of this result. Conjecture 1 (Grillet [6J). Every directed graph G contains an independent set S such that A(G-S) < A(G). A path JIt = (Xl, ... , Xk) will always be a directed and elementary path; it is a longest path if k is maximum, and a non-augmentable path iffor every vertex a, none of the sequences are paths. The anti-path of JIt is the sequence JIt