5 research outputs found

    Visibility Representations of Boxes in 2.5 Dimensions

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    We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane z=0z=0 and edges are unobstructed lines of sight parallel to the xx- or yy-axis. We prove that: (i)(i) Every complete bipartite graph admits a 2.5D-BR; (ii)(ii) The complete graph KnK_n admits a 2.5D-BR if and only if n≤19n \leq 19; (iii)(iii) Every graph with pathwidth at most 77 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an nn-vertex graph that admits a 2.5D-GBR has at most 4n−6n4n - 6 \sqrt{n} edges and this bound is tight. Finally, we prove that deciding whether a given graph GG admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR Γ\Gamma is the set of bottom faces of the boxes in Γ\Gamma.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Dynamic Hierarchical Graph Drawing

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    Area Requirement of Visibility Representations of Trees

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    Area Requirement of Visibility Representations of Trees

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    Abstract We study the area requirement of bar-visibility and rectangle-visibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give linear-time algorithms that construct representations with asymptotically optimal area. 1 Introduction Visibility is a fundamental relation in computational geometry (see, e.g., [1, 8, 9]). In particular, the problem of constructing visibility representations of graphs, where the vertices are drawn as geometric objects of a certain class and the edges are associated with pairs of "visible " objects, has been extensively investigated (see, e.g., [3, 5, 12])
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