Semikernels modulo F and kernels in digraphs Original Research Article

Let D be a finite digraph, V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A kernel N of D is an independent set of vertices such that for every w∈V(D)−N there exists an arc from w to N. Let F be a set of arcs of D (i.e. F⊆A(D)), a set S⊆V(D) is called a semikernel of D modulo F if S is an independent set of vertices such that for every z∈V(D)−S for which there exists an Sz-arc of D−F, there also exists a zS-arc in D. In this paper is introduced the concept of semikernel modulo F and it is used to obtain a new sufficient condition for the existence of kernels in digraphs. As a consequence is obtained a generalization of the following result of B. Sands, N. Sauer and R. Woodrow; in case that the digraph is a finite digraph: Let D be a digraph whose arcs are coloured with two colors. If D contains no monochromatic infinite outward path, then there exists a set S of vertices of D such that: No two vertices of S are connected by a monochromatic directed path and for every vertex not in S there is a monochromatic directed path from x to a vertex in S.