H-kernels by walks in an () digraph

Let be a digraph possibly with loops and a digraph without loops whose arcs are colored with the vertices of ( is said to be an -colored digraph). A directed walk in is said to be an -walk if and only if the consecutive colors encountered on form a directed walk in . A subset of the vertices of is said to be an -kernel by walks if (1) for every pair of different vertices in there is no -walk between them and (2) for each vertex in ()- there exists an -walk from to in .Let be a digraph and suppose that is a digraph possibly infinite. In this paper we will work with the digraph (), where () is an -colored digraph defined as follows: ( ()) = () () and ( ()) = () {(, ) : = (, ) ()} {(, ) : = (, ) ()}, where (, , ) is an -walk in () for every = (, ) in () and every arc of in () is colored with a vertex of . We will show sufficient conditions on both and () in order to guarantee the existence or uniqueness of -kernels by walks in ().