For a digraph D, V (D) and A(D) will denote the sets of vertices and arcs of D respectively. In an arc-colored digraph, a subset K of V(D) is said to be kernel by monochromatic paths (mp-kernel) if (1) for any two different vertices x, y in N there is no monochromatic directed path between them (N is mp-independent) and (2) for each vertex u in V (D) N there exists v ∈ 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 CC(D). In this paper we will approach an mp-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 CC(D) is not strong, then D has an mp-kernel. Previous interesting results are generalized, as for example Richardson′s Theorem.
@article{bwmeta1.element.doi-10_7151_dmgt_1860,
author = {Hortensia Galeana-\'Sanchez and Roc\'\i o S\'anchez-L\'opez},
title = {Kernels by Monochromatic Paths and Color-Perfect Digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {309-321},
zbl = {1338.05100},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1860}
}
Hortensia Galeana-Śanchez; Rocío Sánchez-López. Kernels by Monochromatic Paths and Color-Perfect Digraphs. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 309-321. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1860/