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Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

Editor-in-Chief: Borowiecki, Mieczyslaw

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Kernels by Monochromatic Paths and Color-Perfect Digraphs

Hortensia Galeana-Śanchez / Rocío Sánchez-López
Published Online: 2016-04-15 | DOI: https://doi.org/10.7151/dmgt.1860


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.

Keywords: kernel; kernel perfect digraph; kernel by monochromatic paths; color-class digraph; quasi color-perfect digraph; color-perfect digraph


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About the article

Received: 2014-08-18

Revised: 2015-06-23

Accepted: 2015-06-23

Published Online: 2016-04-15

Published in Print: 2016-05-01

Citation Information: Discussiones Mathematicae Graph Theory, Volume 36, Issue 2, Pages 309–321, ISSN (Online) 2083-5892, DOI: https://doi.org/10.7151/dmgt.1860.

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© by Hortensia Galeana-Śanchez. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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