Richardson’s Theorem in H-coloured Digraphs

Abstract

Let H be a digraph possibly with loops and D a finite digraph without loops whose arcs are coloured with the vertices of H (D is an H -coloured digraph). The sets V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A directed path W in D is an H -path if and only if the consecutive colors encountered on W form a directed walk in H . A set N ⊆ V(D) is an H -kernel if for every pair of different vertices in N there is no H -path between them, and for every vertex u ∈ V(D)\N there exists an H -path in D from u to N . Let D be an m-coloured digraph. The color-class digraph of D, denoted by C C (D), is the digraph such that: the vertices of the color-class digraph are the colors represented in the arcs of D, and (i, j) ∈ A(C C (D)) if and only if there exist two arcs namely (u, v) and (v, w) in D such that (u, v) has color i and (v, w) has color j. Let W = (v 0 , . . . , v n ) be a directed walk in C C (D), with D an H -coloured digraph, and e i = (v i , v i+1 ) for each i ∈ {0, . . . , n − 1}. Let I = {i 1 , . . . , i k } a subset of {0, . . . , n − 1} such that for 0 ≤ s ≤ n − 1, e s ∈ A(H c ) if and only if s ∈ I (where H c is the complement of H ), then we will say that k is the H c -length of W . Since V(C C (D)) ⊆ V(H ), the main question is: What structural properties of C C (D), with respect to H , imply that D has an H -kernel? In this paper we will prove the following: If C C (D) does not have directed cycles of odd H c -length, then D has an H -kernel. Finally we will prove Richardson's theorem as a direct consequence of the previous result.

On kernels in strongly game-perfect digraphs and a characterisation of weakly game-perfect digraphs

“…When one considers digraph classes, the kernel existence problem is a widely-used measure of the complexity of a given class of digraphs [7][8][9]. Recall that a kernel of a digraph D = (V, A) is an independent and absorbing set K , i.e.…”

H-kernels and H-obstructions in H-colored digraphs

“…In [13] the authors also proved the following: Let H be a digraph, D an H-colored digraph and C C (D) its color-class digraph. If every cycle of C C (D) has an even number of arcs in H c , then C H (D) is a kernel perfect digraph.…”

“…Previously we mentioned that Theorem 2.9 was proved in [13]. In order to prove this theorem as a direct consequence of Theorem 2.5 we will need a definition and a Lemma.…”

“…The following Theorem, which was proved in [13], will be useful in order to prove the main results (Theorem 2.5).…”

-panchromatic digraphs

“…Several interesting kinds of kernels are particular cases of H-kernels, as example, kernels by paths, kernels by monochromatic paths, kernels by alternating paths, kernels by rainbow paths, and usual kernels. Several conditions on the existence of H-kernels have been showed, as example, see [19] and [20].…”