Monochromatic kernel-perfectness of special classes of digraphs

A digraph D is said to be an m-coloured digraph, if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) of vertices of D is said to be a kernel by monochromatic paths of the m-coloured digraph D, if it satisfies the two following properties: (1) N is independent by monochromatic paths; i.e., for any two different vertices x, y ∈N , there is no monochromatic directed path between them, and (2) N is absorbent by monochromatic paths; i.e., for each u∈ (V (D) − N) there exists a uv-monochromatic directed path, for some v∈N . In this paper we prove that if D is a digraph without monochromatic directed cycles then: (i) D has a kernel by monochromatic paths iff any composition over D of a family of digraphs (αv)v∈V (D) each one of them having a kernel by monochromatic paths, has a kernel by monochromatic paths, and (ii) D has a kernel by monochromatic paths iff for any B ⊆ V (D) the duplication of D over B, D, has a kernel by monochromatic paths. Also we introduce the concept of locally monochromatic kernel-perfect digraph. As a consequence of the results (i) and (ii) it is proved that: If D is a digraph without monochromatic directed cycles, then: D and each αv, v∈V (D) are locally monochromatic kernel-perfect digraphs iff the composition over D of (αv)v∈V (D) is a locally monochromatic kernel-perfect digraph. And, D is a locally monochromatic kernel-perfect digraph iff for any B ⊆ V (D), the duplication of D over B, D, is a locally monochromatic kernel-perfect digraph.