Discussiones
Mathematicae Graph Theory 20(2) (2000) 243-254
DOI: https://doi.org/10.7151/dmgt.1123
KERNELS IN THE CLOSURE OF COLOURED DIGRAPHS
Hortensia Galeana-Sánchez Instituto de Matemáticas, UNAM |
José de Jesús García-Ruvalcaba Facultad de Ciencias |
Abstract
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.
Keywords and phrases: kernel, closure, tournament.
2000 Mathematics Subject Classification: 05C20.
References
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Received 31 January 2000
Revised 2 October 2000
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