CKI-Digraphs, Generalized Sums and Partitions of Digraphs

Abstract

A kernel of a digraph is a set of vertices which is both independent and absorbent. Let $$D$$D be a digraph such that every proper induced subdigraph of $$D$$D has a kernel; $$D$$D is said to be kernel perfect digraph (KP-digraph) if the digraph $$D$$D has a kernel and critical kernel imperfect digraph (CKI-digraph) if the digraph $$D$$D does not have a kernel. We characterize the CKI-digraphs with a partition into an independent set and a semicomplete digraph. The generalized sum $$G(F_u)$$G(Fu) of a family of mutually disjoint digraphs $$\{F_u\}_{u\in V(G)}$${Fu}u?V(G) over a graph $$G$$G is a digraph defined as follows: Consider $$\cup _{u\in V(G)}F_u$$?u?V(G)Fu, and for each $$x\in V(F_v)$$x?V(Fv) and $$y\in V(F_w)$$y?V(Fw) with $$\{v,w\}\in E(G)$${v,w}?E(G) choose exactly one of the two arcs $$(x,y)$$(x,y) or $$(y,x)$$(y,x). We characterize the asymmetric CKI-digraphs which are generalized sums over an edge or a cycle. Furthermore, we give sufficient conditions on $$G$$G and the family $$\{F_u\}_{u\in V(G)}$${Fu}u?V(G), such that the generalized sum $$G(F_u)$$G(Fu) has a kernel or is a is KP-digraph.