Kernels in the closure of coloured digraphs

Let D be a digraph with V (D) and A(D) the sets of vertices and arcs of D, respectively. A kernel of D is a set I ⊂ V (D) such that no arc of D joins two vertices of I and for each x ∈ V (D) \ I there is a vertex y ∈ I such that (x, y) ∈ A(D). A digraph is kernel-perfect if every non-empty induced subdigraph of D has a kernel. If D is edge coloured, we define the closure ξ(D) of D the multidigraph with V (ξ(D)) = V (D) and A (ξ(D)) = ⋃ i{(u, v) with colour i : there exists a monochromatic path of colour i from the vertex u to the vertex v contained in D}. Let T3 and C3 denote the transitive tournament of order 3 and the 3-cycle, respectively, both of whose arcs are coloured with 3 different colours. In this paper, we survey sufficient conditions for the existence of kernels in the closure of edge coloured digraphs, also we prove that if D is obtained from an edge coloured tournament by deleting one arc and D does not contain T3 or C3, then ξ(D) is a kernel-perfect digraph.