(H, k)-reachability in H-arc-colored digraphs

Let H be a digraph possibly with loops, D be a digraph, and k be an integer, k >= 3. An H-coloring zeta is a map zeta : A(D) -> V (H). An (H, k)-walk W in D is a walk W = (x(0), ... , x(n)) with length at most k such that (zeta(x(0), x(1)), ... , zeta (x(n-1), x(n))) is a walk in H. An (H, k)-path in D is an (H, k)-walk which is a path in D. In this work, we introduce the reachability by (H, k)-paths as follows, for u, v is an element of V (D), we say that u reaches v by (H, k)-paths if there exists an (H, k)-path from u to v in D. Naturally, this new reachability concept can be used to model several connectivity problems. We focus on one of the many aspects of the reachability by (H, k)-paths, the (H, k)- kernels. A subset N of V (D) is an (H, k)-kernel if N is an (H, k)-independent (a subset S of V (D) such that no vertex in S can reach another (different) vertex in S by (H, k - 1)-paths) and (H, k - 1)-absorbent (a subset S of V (D) such that every vertex in V (D) - S reaches some vertex in S by (H, k - 1)-paths). A digraph D is (H, k)-path-quasi-transitive, if for every three vertices x, y and w of D such that there are an (H, k)-path from x to y and an (H, k)-path from y to w in D, then there is an (H, k)-path from x to w or an (H, k)-path from w to x in D. We give sufficient conditions for a (H, k - 1)-path-quasi-transitive digraph that has an (H, k)-kernel. As a main result, we give sufficient conditions for a partition xi of V (H) such that the arc set colored with the colors for every part of xi induces an (H, k - 1)-path-quasi-transitive digraph in D, to imply the existence of an (H, k)-kernel in D. This result generalizes the results of Casas-Bautista et al. (2015), and Hern & aacute;ndez-Lorenzana and S & aacute;nchez-L & oacute;pez (2022). Finally, we show two applications of (H, k)-kernels.