1,248 research outputs found

    Relating Graph Thickness to Planar Layers and Bend Complexity

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    The thickness of a graph G=(V,E)G=(V,E) with nn vertices is the minimum number of planar subgraphs of GG whose union is GG. A polyline drawing of GG in R2\mathbb{R}^2 is a drawing Γ\Gamma of GG, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Γ\Gamma is the maximum number of bends per edge in Γ\Gamma, and the layer complexity of Γ\Gamma is the minimum integer rr such that the set of polygonal chains in Γ\Gamma can be partitioned into rr disjoint sets, where each set corresponds to a planar polyline drawing. Let GG be a graph of thickness tt. By F\'{a}ry's theorem, if t=1t=1, then GG can be drawn on a single layer with bend complexity 00. A few extensions to higher thickness are known, e.g., if t=2t=2 (resp., t>2t>2), then GG can be drawn on tt layers with bend complexity 2 (resp., 3n+O(1)3n+O(1)). However, allowing a higher number of layers may reduce the bend complexity, e.g., complete graphs require Θ(n)\Theta(n) layers to be drawn using 0 bends per edge. In this paper we present an elegant extension of F\'{a}ry's theorem to draw graphs of thickness t>2t>2. We first prove that thickness-tt graphs can be drawn on tt layers with 2.25n+O(1)2.25n+O(1) bends per edge. We then develop another technique to draw thickness-tt graphs on tt layers with bend complexity, i.e., O(2t⋅n1−(1/β))O(\sqrt{2}^{t} \cdot n^{1-(1/\beta)}), where β=2⌈(t−2)/2⌉\beta = 2^{\lceil (t-2)/2 \rceil }. Previously, the bend complexity was not known to be sublinear for t>2t>2. Finally, we show that graphs with linear arboricity kk can be drawn on kk layers with bend complexity 3(k−1)n(4k−2)\frac{3(k-1)n}{(4k-2)}.Comment: A preliminary version appeared at the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016

    Single failure resiliency in greedy routing

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    Using greedy routing, network nodes forward packets towards neighbors which are closer to their destination. This approach makes greedy routers significantly more memory-efficient than traditional IP-routers using longest-prefix matching. Greedy embeddings map network nodes to coordinates, such that greedy routing always leads to the destination. Prior works showed that using a spanning tree of the network topology, greedy embeddings can be found in different metric spaces for any graph. However, a single link/node failure might affect the greedy embedding and causes the packets to reach a dead end. In order to cope with network failures, existing greedy methods require large resources and cause significant loss in the quality of the routing (stretch loss). We propose efficient recovery techniques which require very limited resources with minor effect on the stretch. As the proposed techniques are protection, the switch-over takes place very fast. Low overhead, simplicity and scalability of the methods make them suitable for large-scale networks. The proposed schemes are validated on large topologies with properties similar to the Internet. The performances of the schemes are compared with an existing alternative referred as gravity pressure routing

    On a Tree and a Path with no Geometric Simultaneous Embedding

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    Two graphs G1=(V,E1)G_1=(V,E_1) and G2=(V,E2)G_2=(V,E_2) admit a geometric simultaneous embedding if there exists a set of points P and a bijection M: P -> V that induce planar straight-line embeddings both for G1G_1 and for G2G_2. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has depth 4.Comment: 42 pages, 33 figure
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