Restricted domination in arc-colored digraphs

Let H = (V (H), A(H)) be a digraph possibly with loops and D = (V (D), A(D)) a digraph whose arcs are colored with the vertices of H (this is what we call an H-colored digraph); i.e. there exists a function c : A(D) → V (H); for an arc of D, f = (u, v) ∈ A(D), we call c(f) = c(u, v) the color of f . A directed walk (directed path) P = (u0, u1, . . . , un) in D will be called an H-walk (H-path) whenever (c(u0, u1), c(u1, u2), . . . , c(un−2, un−1), c(un−1, un)) is a directed walk (directed path) in H. We introduce the concept of H-kernel N , as a generalization of the two properties that define a kernel (Recall that a kernel N of a digraph D is a set of vertices N ⊆ V (D) which is independent and for each x ∈ V (D) − N , there exists an xN -arc in D). A set N ⊆ V (D) is called H-independent whenever for every two different vertices x, y ∈ N there is no H-path between them, and N is called H-absorbent whenever for each x ∈ V (D)−N there exists a vertex y ∈ N and an xy-H-path in D. The set N ⊆ V (D) will be called H-kernel if and only if it is H-independent and H-absorbent. This new concept generalizes the concepts of kernel, kernel by monochromatic paths and kernel by alternating paths. In this paper we show sufficient conditions for an infinite digraph to have an H-kernel.