9,365 research outputs found

    A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings

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    A plus-contact representation of a planar graph GG is called cc-balanced if for every plus shape +v+_v, the number of other plus shapes incident to each arm of +v+_v is at most cΔ+O(1) c \Delta +O(1), where Δ\Delta is the maximum degree of GG. Although small values of cc have been achieved for a few subclasses of planar graphs (e.g., 22- and 33-trees), it is unknown whether cc-balanced representations with c<1c<1 exist for arbitrary planar graphs. In this paper we compute (1/2)(1/2)-balanced plus-contact representations for all planar graphs that admit a rectangular dual. Our result implies that any graph with a rectangular dual has a 1-bend box-orthogonal drawings such that for each vertex vv, the box representing vv is a square of side length deg(v)2+O(1)\frac{deg(v)}{2}+ O(1).Comment: A poster related to this research appeared at the 25th International Symposium on Graph Drawing & Network Visualization (GD 2017

    Contact Representations of Graphs in 3D

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    We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

    Drawing a Graph in a Hypercube

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    A dd-dimensional hypercube drawing of a graph represents the vertices by distinct points in {0,1}d\{0,1\}^d, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.Comment: Submitte

    The DFS-heuristic for orthogonal graph drawing☆☆Some of these result were published in the author's PhD thesis at Rutgers University; the author would like to thank her advisor, Prof. Endre Boros, for much helpful input. The results in Section 5 have been presented at the 8th Canadian Conference on Computational Geometry, Ottawa, 1996, see [1].

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    AbstractIn this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of m−2 bends

    Quantum automorphism groups of homogeneous graphs

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    Associated to a finite graph XX is its quantum automorphism group GG. The main problem is to compute the Poincar\'e series of GG, meaning the series f(z)=1+c1z+c2z2+...f(z)=1+c_1z+c_2z^2+... whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives ff for all homogeneous graphs having 8 vertices or less.Comment: 30 page
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