689 research outputs found

    Stack and Queue Layouts via Layered Separators

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    It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed gg and kk, we show that every nn-vertex graph that can be embedded on a surface of genus gg with at most kk crossings per edge has stack-number O(logn)\mathcal{O}(\log n); this includes kk-planar graphs. The previously best known bound for the stack-number of these families was O(n)\mathcal{O}(\sqrt{n}), except in the case of 11-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that nn-vertex graphs that admit constant layered separators have O(logn)\mathcal{O}(\log n) stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs

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    We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth

    The degree-diameter problem for sparse graph classes

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    The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree Δ\Delta and diameter kk. For fixed kk, the answer is Θ(Δk)\Theta(\Delta^k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(Δk1)\Theta(\Delta^{k-1}), and for graphs of bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases for fixed kk. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs

    Clustered 3-Colouring Graphs of Bounded Degree

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    A (not necessarily proper) vertex colouring of a graph has "clustering" cc if every monochromatic component has at most cc vertices. We prove that planar graphs with maximum degree Δ\Delta are 3-colourable with clustering O(Δ2)O(\Delta^2). The previous best bound was O(Δ37)O(\Delta^{37}). This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree Δ\Delta that exclude a fixed minor are 3-colourable with clustering O(Δ5)O(\Delta^5). The best previous bound for this result was exponential in Δ\Delta.Comment: arXiv admin note: text overlap with arXiv:1904.0479
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