168 research outputs found

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

    Full text link
    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

    Hitting minors, subdivisions, and immersions in tournaments

    Full text link
    The Erd\H{o}s-P\'osa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph HH (resp. strongly-connected directed graph HH), the class of directed graphs that contain HH as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove that if HH is a strongly-connected directed graph, the class of directed graphs containing HH as an immersion has the edge-Erd\H{o}s-P\'osa property in the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science. Difference with the previous version: use of the DMTCS article class. For a version with hyperlinks see the previous versio

    A classification of locally semicomplete digraphs

    Get PDF
    Recently, Huang (1995) gave a characterization of local tournaments. His characterization involves arc-reversals and therefore may not be easily used to solve other structural problems on locally semicomplete digraphs (where one deals with a fixed locally semicomplete digraph). In this paper we derive a classification of locally semicomplete digraphs which is very useful for studying structural properties of locally semicomplete digraphs and which does not depend on Huang's characterization. An advantage of this new classification of locally semicomplete digraphs is that it allows one to prove results for locally semicomplete digraphs without reproving the same statement for tournaments. We use our result to characterize pancyclic and vertex pancyclic locally semicomplete digraphs and to show the existence of a polynomial algorithm to decide whether a given locally semicomplete digraph has a kernel

    On the pathwidth of almost semicomplete digraphs

    Full text link
    We call a digraph {\em hh-semicomplete} if each vertex of the digraph has at most hh non-neighbors, where a non-neighbor of a vertex vv is a vertex u≠vu \neq v such that there is no edge between uu and vv in either direction. This notion generalizes that of semicomplete digraphs which are 00-semicomplete and tournaments which are semicomplete and have no anti-parallel pairs of edges. Our results in this paper are as follows. (1) We give an algorithm which, given an hh-semicomplete digraph GG on nn vertices and a positive integer kk, in (h+2k+1)2knO(1)(h + 2k + 1)^{2k} n^{O(1)} time either constructs a path-decomposition of GG of width at most kk or concludes correctly that the pathwidth of GG is larger than kk. (2) We show that there is a function f(k,h)f(k, h) such that every hh-semicomplete digraph of pathwidth at least f(k,h)f(k, h) has a semicomplete subgraph of pathwidth at least kk. One consequence of these results is that the problem of deciding if a fixed digraph HH is topologically contained in a given hh-semicomplete digraph GG admits a polynomial-time algorithm for fixed hh.Comment: 33pages, a shorter version to appear in ESA 201
    • …
    corecore