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. ...
