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

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 ( 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 obstruction on v i if ( c ( v i - 1 , v i ), c ( v i , v i + 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.