A sufficient condition for an infinite digraph to be kernel-perfect Hortensia Galeana-S´ anchez a , Mucuy-kak Guevara a a Instituto de Matem´ aticas Universidad Nacional Aut´ onoma de M´ exico Ciudad Universitaria, M´ exico D.F, 04510, M´ exico Abstract In this paper we show a sufficient condition for an infinite digraph to be kernel- perfect, as a consequence we obtain several generalizations of well known results on the existence of kernels in finite and infinite digraphs. Key words: infinite digraphs, kernel, semikernel modulo F, kernel-perfect digraph, critical kernel imperfect digraph. 1 Introduction In this paper D will denote a loopless infinite digraph, unless the contrary be said, with possibly multiple arcs; V (D) and A(D) will denote the sets of vertices and arcs of D respectively. An arc u 1 u 2 ∈ A(D) is called asymmetrical (resp symmetrical) if u 2 u 1 / ∈ A(D) (resp. u 2 u 1 ∈ A(D)). The asymmetrical (resp. symmetrical) part of D which is denoted by Asym(D) (resp. Sym(D)) is the spanning subdigraph of D whose arcs are the asymmetrical (resp. symmet- rical) arcs of D. We recall that a subdigraph D 1 of D is a spanning subdigraph if V (D 1 )= V (D). If S is a nonempty set of V (D) then the subdigraph D[S ] induced by S is the digraph with vertex the set S and those arcs of D which join vertices of S . An arc u 1 u 2 of D will be called and (S 1 ,S 2 )-arc whenever u 1 ∈ S 1 and u 2 ∈ S 2 . A directed path is a finite sequence x 1 ,x 2 ,...,x n of distinct vertices of D such that x i x i+1 ∈ A(D) for each 1 ≤ i ≤ n - 1. When the sequence is infinite we Email addresses: hgaleana@matem.unam.mx (Hortensia Galeana-S´ anchez), guevara@matem.unam.mx (Mucuy-kak Guevara). Preprint submitted to Elsevier Science 21 February 2006