For a path system P=(x,R,S,T), where S⊆ X, T ⊆ X, and R⊆ X × X × X, the problem of whether there is an admissible vertex in S. A vertex is admissible if and only if x∈ T, or there exists admissible y, z ∈ X such that (x,y,z) ∈ R.