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