Kernels in infinite digraphs

Abstract

Let D be a digraph, possibly infinite, V(D) and A(D) 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. A set S ⊆ V (D) is a semikernel of D if it is independent and (u, v) ∊ A(D) with u ∊ S implies that there is an arc from v to some vertex in S. In this paper we introduce sufficient conditions for the existence of kernels in some kinds of infinite digraphs, such as transitive digraphs, symmetric digraphs, acyclic digraphs and digraphs without odd cycles. We use strongly the concept of semikernel.