4,712 research outputs found

    The Generalised Colouring Numbers on Classes of Bounded Expansion

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    The generalised colouring numbers admr(G)\mathrm{adm}_r(G), colr(G)\mathrm{col}_r(G), and wcolr(G)\mathrm{wcol}_r(G) were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of rr. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors

    An FPT Algorithm for Directed Spanning k-Leaf

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    An out-branching of a directed graph is a rooted spanning tree with all arcs directed outwards from the root. We consider the problem of deciding whether a given directed graph D has an out-branching with at least k leaves (Directed Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when k is chosen as the parameter. Previously this was only known for restricted classes of directed graphs. The main new ingredient in our approach is a lemma that shows that given a locally optimal out-branching of a directed graph in which every arc is part of at least one out-branching, either an out-branching with at least k leaves exists, or a path decomposition with width O(k^3) can be found. This enables a dynamic programming based algorithm of running time 2^{O(k^3 \log k)} n^{O(1)}, where n=|V(D)|.Comment: 17 pages, 8 figure

    Spanning trees with few branch vertices

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    A branch vertex in a tree is a vertex of degree at least three. We prove that, for all s1s\geq 1, every connected graph on nn vertices with minimum degree at least (1s+3+o(1))n(\frac{1}{s+3}+o(1))n contains a spanning tree having at most ss branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat

    On Feedback Vertex Set: New Measure and New Structures

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    We present a new parameterized algorithm for the {feedback vertex set} problem ({\sc fvs}) on undirected graphs. We approach the problem by considering a variation of it, the {disjoint feedback vertex set} problem ({\sc disjoint-fvs}), which finds a feedback vertex set of size kk that has no overlap with a given feedback vertex set FF of the graph GG. We develop an improved kernelization algorithm for {\sc disjoint-fvs} and show that {\sc disjoint-fvs} can be solved in polynomial time when all vertices in GFG \setminus F have degrees upper bounded by three. We then propose a new branch-and-search process on {\sc disjoint-fvs}, and introduce a new branch-and-search measure. The process effectively reduces a given graph to a graph on which {\sc disjoint-fvs} becomes polynomial-time solvable, and the new measure more accurately evaluates the efficiency of the process. These algorithmic and combinatorial studies enable us to develop an O(3.83k)O^*(3.83^k)-time parameterized algorithm for the general {\sc fvs} problem, improving all previous algorithms for the problem.Comment: Final version, to appear in Algorithmic
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