Kernels and some operations in edge-coloured digraphs

Let DD be an edge-coloured digraph, V(D)V(D) will denote the set of vertices of DD; a set N⊆V(D)N⊆V(D) is said to be a kernel by monochromatic paths of DD if it satisfies the following two conditions: For every pair of different vertices u,v∈Nu,v∈N there is no monochromatic directed path between them and; for every vertex x∈V(D)−Nx∈V(D)−N there is a vertex y∈Ny∈N such that there is an xyxy-monochromatic directed path.In this paper we consider some operations on edge-coloured digraphs, and some sufficient conditions for the existence or uniqueness of kernels by monochromatic paths of edge-coloured digraphs formed by these operations from another edge-coloured digraphs.