1,829 research outputs found

    An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

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    Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an nn-vertex graph GG and a positive integer kk, we want to decide whether there is a set of at most kk vertices whose removal turns GG into a graph of linear rankwidth at most 11 and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time f(k)⋅n3f(k)\cdot n^3 for some function ff, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time 8k⋅nO(1)8^k\cdot n^{\mathcal{O}(1)}. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time 2O(k)⋅n42^{\mathcal{O}(k)}\cdot n^4. We also prove that the running time cannot be improved to 2o(k)⋅nO(1)2^{o(k)}\cdot n^{\mathcal{O}(1)} under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201

    Optimal path and cycle decompositions of dense quasirandom graphs

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    Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<10<p<1 be constant and let G∌Gn,pG\sim G_{n,p}. Let odd(G)odd(G) be the number of odd degree vertices in GG. Then a.a.s. the following hold: (i) GG can be decomposed into ⌊Δ(G)/2⌋\lfloor\Delta(G)/2\rfloor cycles and a matching of size odd(G)/2odd(G)/2. (ii) GG can be decomposed into max⁥{odd(G)/2,⌈Δ(G)/2⌉}\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\} paths. (iii) GG can be decomposed into ⌈Δ(G)/2⌉\lceil\Delta(G)/2\rceil linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte

    Completing Partial Packings of Bipartite Graphs

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    Given a bipartite graph HH and an integer nn, let f(n;H)f(n;H) be the smallest integer such that, any set of edge disjoint copies of HH on nn vertices, can be extended to an HH-design on at most n+f(n;H)n+f(n;H) vertices. We establish tight bounds for the growth of f(n;H)f(n;H) as n→∞n \rightarrow \infty. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that f(n;H)=o(n)f(n;H) = o(n). This settles a long-standing open problem

    Linear kernels for outbranching problems in sparse digraphs

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    In the kk-Leaf Out-Branching and kk-Internal Out-Branching problems we are given a directed graph DD with a designated root rr and a nonnegative integer kk. The question is to determine the existence of an outbranching rooted at rr that has at least kk leaves, or at least kk internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k2)O(k^2) vertices are known on general graphs. In this work we show that kk-Leaf Out-Branching admits a kernel with O(k)O(k) vertices on H\mathcal{H}-minor-free graphs, for any fixed family of graphs H\mathcal{H}, whereas kk-Internal Out-Branching admits a kernel with O(k)O(k) vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page

    Rainbow Hamilton cycles in random regular graphs

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    A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page
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