648 research outputs found

    Embedding Four-directional Paths on Convex Point Sets

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    A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{nn-vertex} four-directional path PP on a set SS of nn points in the plane is a straight-line drawing of PP where each vertex of PP is mapped to a distinct point of SS and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.Comment: 11 pages, full conference version including all proof

    Upward Point-Set Embeddability

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    We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph DD has an upward planar embedding into a point set SS. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of kk-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a 11-switch tree), we show that not every kk-switch tree admits an upward planar straight-line embedding into any convex point set, for any k2k \geq 2. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete

    Small Superpatterns for Dominance Drawing

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    We exploit the connection between dominance drawings of directed acyclic graphs and permutations, in both directions, to provide improved bounds on the size of universal point sets for certain types of dominance drawing and on superpatterns for certain natural classes of permutations. In particular we show that there exist universal point sets for dominance drawings of the Hasse diagrams of width-two partial orders of size O(n^{3/2}), universal point sets for dominance drawings of st-outerplanar graphs of size O(n\log n), and universal point sets for dominance drawings of directed trees of size O(n^2). We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}), riffle permutations (321-, 2143-, and 2413-avoiding permutations) have superpatterns of size O(n), and the concatenations of sequences of riffles and their inverses have superpatterns of size O(n\log n). Our analysis includes a calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of the 321-superpattern siz

    The Partial Visibility Representation Extension Problem

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    For a graph GG, a function ψ\psi is called a \emph{bar visibility representation} of GG when for each vertex vV(G)v \in V(G), ψ(v)\psi(v) is a horizontal line segment (\emph{bar}) and uvE(G)uv \in E(G) iff there is an unobstructed, vertical, ε\varepsilon-wide line of sight between ψ(u)\psi(u) and ψ(v)\psi(v). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph GG, a bar visibility representation ψ\psi of GG, additionally, puts the bar ψ(u)\psi(u) strictly below the bar ψ(v)\psi(v) for each directed edge (u,v)(u,v) of GG. We study a generalization of the recognition problem where a function ψ\psi' defined on a subset VV' of V(G)V(G) is given and the question is whether there is a bar visibility representation ψ\psi of GG with ψ(v)=ψ(v)\psi(v) = \psi'(v) for every vVv \in V'. We show that for undirected graphs this problem together with closely related problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

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    When can a plane graph with prescribed edge lengths and prescribed angles (from among {0,180,360\{0,180^\circ, 360^\circ\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to 360360^\circ, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

    Strictly convex drawings of planar graphs

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    Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma

    Planarity Variants for Directed Graphs

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    Combinatorial and Geometric Aspects of Computational Network Construction - Algorithms and Complexity

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    Tight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography

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    We study maximal identifiability, a measure recently introduced in Boolean Network Tomography to characterize networks' capability to localize failure nodes in end-to-end path measurements. We prove tight upper and lower bounds on the maximal identifiability of failure nodes for specific classes of network topologies, such as trees and dd-dimensional grids, in both directed and undirected cases. We prove that directed dd-dimensional grids with support nn have maximal identifiability dd using 2d(n1)+22d(n-1)+2 monitors; and in the undirected case we show that 2d2d monitors suffice to get identifiability of d1d-1. We then study identifiability under embeddings: we establish relations between maximal identifiability, embeddability and graph dimension when network topologies are model as DAGs. Our results suggest the design of networks over NN nodes with maximal identifiability Ω(logN)\Omega(\log N) using O(logN)O(\log N) monitors and a heuristic to boost maximal identifiability on a given network by simulating dd-dimensional grids. We provide positive evidence of this heuristic through data extracted by exact computation of maximal identifiability on examples of small real networks
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