A Richardson's theorem version for ⊓-kernels

Abstract Let D = ( V ( D ) , A ( D ) ) be a digraph, D P ( D ) be the set of directed paths of D and let Π be a subset of D P ( D ) . A subset S ⊆ V ( D ) will be called Π -independent if for any pair { x , y } ⊆ S , there is no x y -path nor y x -path in Π ; and will be called Π -absorbing if for every x ∈ V ( D ) ∖ S there is y ∈ S such that there is an x y -path in Π . A set S ⊆ V ( D ) will be called a Π -kernel if S is Π -independent and Π -absorbing. Given x , y ∈ V ( D ) and Q ⊆ Π we will use ( x → Q y ) (respectively ( x ↛ Q y ) ) to denote the fact that there is an (respectively, no) x y -path in Q . A subset Q ⊆ Π will be called transitive if whenever ( x → Q y ) and ( y → Q z ) , then ( x → Q z ) . Given a partition P = { Π i } i ∈ I of Π , the color digraph of D and I is the digraph C D ( P ) with I as vertex-set and given any pair { j , k } ⊆ I , the arc j k belongs to C D ( P ) if and only if there are x , y , w ∈ V ( D ) such that ( x → Π j y ) and ( y → Π k w ) . In this paper we prove the following result: Let D be a digraph and Π ⊆ D P ( D ) . If there is a partition P = { Π i } i ∈ I of Π such that: For each i ∈ I , Π i is transitive; and C D ( P ) has no odd cycles of order greater than 1, then D has Π - kernel. Some interesting previous results are obtained as a direct consequence of this result.