Formula omitted]-kernels and [formula omitted]-obstructions in [formula omitted]-colored digraphs

Abstract

Let 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 (z1, z2) of D we will denote its color by c(z1, z2). We introduce the concept of obstruction in an H-colored digraph as follows. Let W=(v0,v1,…,vn−1,v0) be a closed directed walk in D. We will say that there is an obstruction on vi if (c(vi−1, vi), c(vi, vi+1)) is 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.