1,395 research outputs found

    k-colored kernels

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    We study kk-colored kernels in mm-colored digraphs. An mm-colored digraph DD has kk-colored kernel if there exists a subset KK of its vertices such that (i) from every vertex v∉Kv\notin K there exists an at most kk-colored directed path from vv to a vertex of KK and (ii) for every u,v∈Ku,v\in K there does not exist an at most kk-colored directed path between them. In this paper, we prove that for every integer k≥2k\geq 2 there exists a (k+1)% (k+1)-colored digraph DD without kk-colored kernel and if every directed cycle of an mm-colored digraph is monochromatic, then it has a kk-colored kernel for every positive integer k.k. We obtain the following results for some generalizations of tournaments: (i) mm-colored quasi-transitive and 3-quasi-transitive digraphs have a kk% -colored kernel for every k≥3k\geq 3 and k≥4,k\geq 4, respectively (we conjecture that every mm-colored ll-quasi-transitive digraph has a kk% -colored kernel for every k≥l+1)k\geq l+1), and (ii) mm-colored locally in-tournament (out-tournament, respectively) digraphs have a kk-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most kk-colored

    Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

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    A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is C→3\overrightarrow{C}_3 and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric 33-quasi-transitive and asymmetric 33-anti-quasi-transitive TT3TT_3-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.Comment: 13 pages and 5 figure

    On the existence and number of (k+1)(k+1)-kings in kk-quasi-transitive digraphs

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    Let D=(V(D),A(D))D=(V(D), A(D)) be a digraph and k≥2k \ge 2 an integer. We say that DD is kk-quasi-transitive if for every directed path (v0,v1,...,vk)(v_0, v_1,..., v_k) in DD, then (v0,vk)∈A(D)(v_0, v_k) \in A(D) or (vk,v0)∈A(D)(v_k, v_0) \in A(D). Clearly, a 2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense. Bang-Jensen and Gutin proved that a quasi-transitive digraph DD has a 3-king if and only if DD has a unique initial strong component and, if DD has a 3-king and the unique initial strong component of DD has at least three vertices, then DD has at least three 3-kings. In this paper we prove the following generalization: A kk-quasi-transitive digraph DD has a (k+1)(k+1)-king if and only if DD has a unique initial strong component, and if DD has a (k+1)(k+1)-king then, either all the vertices of the unique initial strong components are (k+1)(k+1)-kings or the number of (k+1)(k+1)-kings in DD is at least (k+2)(k+2).Comment: 17 page
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