h-kernels by walks in subdivision digraph

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