KERNELS BY MONOCHROMATIC PATHS AND COLOR-PERFECT DIGRAPHS

For a digraph D, V(D) and A(D) will denote the vertices and arcs of 1) respectively. In an arc-colored digraph, a subset K of V( D) is said to be kernel by monochromatic paths (mp-kernel) if (1) far any two different vertices x, y in N there is no monochromatic directed path between them (AT is nip-independent) and (2) for each vertex u in V(D) \ N there exists v is an element of N such that there is a monochromatic directed path from u to v in D (N is mp-absorbent). If every arc in D has a different color, then a kernel by monochromatic paths is said to be a kernel. Two associated digraphs to an arc -colored digraph are the closure and the color-class digraph e(c)(D). In this paper we will approach an trip-kernel via the closure of induced subdigraphs of D which have the property of having few colors in their arcs with respect to D. We will introduce the concept of color-perfect digraph and we are going to prove that if D is an arc-colored digraph such that D is a quasi color perfect digraph and e(c)(D) is not strong, then D has an mp-kernel. Previous interesting results are generalized, as for example Richardson's Theorem.