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

Abstract

a b s t r a c tLet D be a digraph. V (D) and A(D) will denote the vertex and arc sets of D respectively. A kernel K of a digraph D is an independent set of vertices of D such that for every vertex w in V (D)-K there exists an arc from w to a vertex in K . Let H be a digraph possibly with loops and D a digraph without loops whose arcs are colored with the vertices of H (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A generalization of the concept of kernel is the concept of H-kernel, where an H-kernel N of an H-colored digraph D is a set of vertices of D such that for every pair of different vertices in N there is no H-path between them, and for every vertex u in V (D)-N there exists an H-path in D from u to N.A classical result in kernel theory establishes that if D is a digraph without cycles of odd length, then D has a kernel; this result is known as Richardson's theorem and in this paper we will show an extension of this theorem which is given by the main result.Let D be an H-colored digraph. For an arc (z 1 , z 2 ) of D we will denote its color by c(z 1 , z 2 ). We introduce the concept of obstruction in an H-colored digraph as follows. Let W = (v 0 , v 1 , . . . , v n−1 , v 0 ) be a closed directed walk in D. We will say that there is an obstructionis not an arc of A(H) (indices modulo n). The main result establishes that if D is an H-colored digraph such that the number of obstructions in every closed directed trail of D is even, then D has an H-kernel. Previous interesting results are generalized, as for example Sands, Sauer and Woodrow's theorem.

Fixed points and connections between positive and negative cycles in Boolean networks

“…Here, the parity of a cycle is the parity of its length. Many generalizations of this results have been established, see for instance [11,12] and the references therein. In [25] a correspondence with Boolean networks shows that Richardson's theorem is a corollary of Theorem 2, and using this correspondence, we deduce from Theorem 5 a new generalization of Richardson's theorem:…”

$(k,H)$-kernels in nearly tournaments

“…If S ⊆ V (D) and x n ∈ S, we say that W is an x 0 S-H-path. In [35] Linek and Sands introduced the concept of H-walk and their work was later considered by several authors, as example, [2,13,22] and [38]. A subset S of vertices of D is absorbent by H. Galeana-Sánchez and M. Tecpa-Galván H-paths, if for every x ∈ V (D) \ S there exists an xS-H-path; and S is said to be independent by H-paths, if for every pair of different vertices {u, v} ⊆ S, there is no uv-H-path between them.…”

-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].…”

“…. , x n ), is an H-walk (H-path), whenever the consecutive colors encountered on W form a directed walk in H. In [22] Linek and Sands introduced the concept of H-walk and their work was later considered by several authors, as example, [2], [8], [16] and [24]. A subset S of vertices of D is absorbent by H-paths (absorbent by H-walks), if for every x ∈ V (D) \ S there exists an xS-H-path (xS-H-walk); and S is said to be independent by H-paths (independent by H-walks), if for every pair of different vertices {u, v} ⊆ S, there is no uv-H-path (uv-H-walk) between them.…”

“…The H-length was primarily studied by Galeana-Sánchez and Sánchez-López in [16] for closed walks, and by Andenmatten, Galeana-Sáchez and Pach in [1] for open paths. Clearly, the usual length l(W ) coincides with the H-length l H (W ), in the very particular case when A(H) = ∅.…”