Structural properties of CKI-digraphs
نویسنده
چکیده
A kernel of a digraph is a set of vertices which is both independent and absorbant. Let D be a digraph such that every proper induced subdigraph has a kernel. If D has a kernel, then D is kernel perfect, otherwise D is critical kernel-imperfect (for short CKI-digraph). In this work we prove that if a CKI-digraph D is not 2-arc connected, then D − a is kernel perfect for any bridge a of D. If D has no kernel but for all vertex x, D − x has a kernel, then D is called kernel critical. We give conditions on a kernel critical digraph D so that for all x ∈ V (D) the kernel of D− x has at least two vertices. Concerning asymmetric digraphs, we show that every vertex u of an asymmetric CKI-digraph D on n ≥ 5 vertices satisfies d(u) + d−(u) ≤ n− 3 and d(u), d−(u) ≤ n− 5. As a consequence, we establish that there are exactly four asymmetric CKI-digraphs on n ≤ 7 vertices. Furthermore, we study the maximum order of a subtournament contained in a not necessarily asymmetric CKI-digraph.
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