Showing papers on "Planarity testing published in 2014"

Form-finding with polyhedral meshes made simple

Abstract: We solve the form-finding problem for polyhedral meshes in a way which combines form, function and fabrication; taking care of user-specified constraints like boundary interpolation, planarity of faces, statics, panel size and shape, enclosed volume, and last, but not least, cost. Our main application is the interactive modeling of meshes for architectural and industrial design. Our approach can be described as guided exploration of the constraint space whose algebraic structure is simplified by introducing auxiliary variables and ensuring that constraints are at most quadratic. Computationally, we perform a projection onto the constraint space which is biased towards low values of an energy which expresses desirable "soft" properties like fairness. We have created a tool which elegantly handles difficult tasks, such as taking boundary-alignment of polyhedral meshes into account, planarization, fairing under planarity side conditions, handling hybrid meshes, and extending the treatment of static equilibrium to shapes which possess overhanging parts.

Non-Rigid Shape-from-Motion for Isometric Surfaces using Infinitesimal Planarity.

Abstract: This paper proposes a general framework to solve Non-Rigid Shape-from-Motion (NRSfM) with the perspective camera under isometric deformations. Contrary to the usual low-rank linear shape basis, isometry allows us to recover complex shape deformations from a sparse set of images. Existing methods suffer from ambiguities and may be very expensive to solve. We bring four main contributions. First, we formulate isometric NRSfM as a system of first-order Partial Differential Equations (PDE) involving the shape’s depth and normal field and an unknown template. Second, we show this system cannot be locally resolved. Third, we introduce the concept of infinitesimal planarity and show that it makes the system locally solvable for at least three views. Fourth, we derive an analytic solution which involves convex, linear least-squares optimization only, and outperforms existing works.

Tight bounds for planar strongly connected steiner subgraph with fixed number of terminals (and extensions)

Abstract: Given a vertex-weighted directed graph G = (V, E) and a set T = {t1, t2, ... tk} of k terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find a vertex set H ⊆ V of minimum weight such that G[H] contains a ti → tj path for each i ≠ j. The problem is NP-hard, but Feldman and Ruhl (FOCS '99; SICOMP '06) gave a novel nO(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs.• Our main algorithmic result is a 2O(k log k) · nO(√k) algorithm for planar SCSS, which is an improvement of a factor of O(√k) in the exponent over the algorithm of Feldman and Ruhl.• Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f(k) · no(√k) algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails.The algorithm eventually relies on the excluded grid theorem for planar graphs, but we stress that it is not simply a straightforward application of treewidth-based techniques: we need several layers of abstraction to arrive to a problem formulation where the speedup due to planarity can be exploited. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a grid-like fashion to tightly control the number of terminals in the created instance.The following additional results put our upper and lower bounds in context:• Our 2O(k log k) · nO(√k) algorithm for planar directed graphs can be generalized to graphs excluding a fixed minor.• In general graphs, we cannot hope for such a dramatic improvement over the nO(k) algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an f(k) · no(k/logk) algorithm for any computable function f.• Feldman and Ruhl generalized their nO(k) algorithm to the more general Directed Steiner Forest (DSF) problem; here the task is to find a subgraph of minimum weight such that for every source si there is a path to the corresponding terminal ti. We show that that, assuming ETH, there is no f(k) · no(k) time algorithm for DSF on acyclic planar graphs.

Graph drawing beyond planarity: some results and open problems.

Abstract: We briefly review some recent findings and outline some emerging research directions about the theory of “nearly planar” graphs, i.e. graphs that have drawings where some crossing configurations are forbidden. 1 Graph drawing beyond planarity Recent technological advances have generated torrents of relational data that are hard to display and visually analyze due, mainly, to their large size. Application domains where this need is particularly pressing include Systems Biology, Social Network Analysis, Software Engineering, and Networking. What is required is not simply an incremental improvement to scale up known solutions but, rather, a quantum jump in the sophistication of the visualization systems and techniques. New research scenarios for visual analytics, network visualization, and human-computer interaction paradigms must be identified; new combinatorial models must be defined and their corresponding theoretical problems must be computationally investigated; finally, the theoretical solutions must be experimentally evaluated and put into practice. Therefore, a substantial research effort in the graph drawing and network visualization communities started from the following considerations. The Planarity Handicap. The classical literature on graph drawing and network visualization showcases elegant algorithms and sophisticated data structures under the assumption that the input relational data set can be displayed as a network where no two edges cross (see, e.g., [14,35,36,40]), i.e. as a planar graph. Unfortunately, almost every graph is non-planar in practice and various experimental studies have established that the human ability of understanding a diagram is dramatically affected by the type and number of edge crossings (see, e.g., [42,43,48]). Combinatorial Topology vs. Algorithmics. A topological graph is a drawing of a graph in the plane such that vertices are drawn as points and edges are drawn as simple arcs between the points. Extremal theory questions such as “how many edges can a certain type of non-planar topological graph have?” have been investigated by mathematicians for decades, typically under the name of Turan-type problems. However, the corresponding computational question: “How efficiently can one compute a drawing Γ of a non-planar graph such that Γ is a topological graph of a certain type?” has been surprisingly disregarded by the algorithmic community until very recent years.

Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth

Abstract: We investigate crossing minimization for 1-page and 2-page book drawings We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth

Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data

Abstract: A method to provide absolute planarity measurements through an interferometric oblique incidence setup and an iterative algorithm is presented. With only three measurements, the calibration of absolute planarity is achieved in a fast and effective manner. Demonstration with synthetic data is provided, and the possible application to very long flat mirrors is pointed out.

On the probability of planarity of a random graph near the critical point

Abstract: Erd˝ os and Rconjectured in 1960 that the limiting probability p that a random graph with n vertices and M = n=2 edges is planar exists. It has been shown that indeed p exists and is a constant strictly between 0 and 1. In this paper we answer completely this long standing question by finding an exact expression for this probability, whose approximate value turns out to bep 0:99780. More generally, we compute the probability of planarity at the critical window of widthn 2=3 around the critical point M = n=2. We extend these results to some classes of graphs closed under taking minors. As an example, we show that the probability of being series-parallel converges to 0:98003. Our proofs rely on exploiting the structure of random graphs in the critical window, obtained previously by Janson, Euczak and Wierman, by means of generating functions and analytic methods. This is a striking example of how analytic combinatorics can be applied to classical problems on random graphs.

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Abstract: We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.

On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs

Abstract: An $st$-path in a drawing of a graph is self-approaching if during the traversal of the corresponding curve from $s$ to any point $t'$ on the curve the distance to $t'$ is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path. We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. We prove that strongly monotone (and thus increasing-chord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. [GD'14]. Moreover, we provide a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.

Clustered Planarity Testing Revisited

Abstract: The Hanani---Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this classical result to clustered graphs with two disjoint clusters, and show that a straightforward extension of our result to flat clustered graphs with three or more disjoint clusters is not possible. We also give a new and short proof for a related result by Di Battista and Frati based on the matroid intersection algorithm.

Morphing Planar Graph Drawings with Unidirectional Moves.

Abstract: Alamdari et al. showed that given two straight-line planar drawings of a graph, there is a morph between them that preserves planarity and consists of a polynomial number of steps where each step is a \emph{linear morph} that moves each vertex at constant speed along a straight line. An important step in their proof consists of converting a \emph{pseudo-morph} (in which contractions are allowed) to a true morph. Here we introduce the notion of \emph{unidirectional morphing} step, where the vertices move along lines that all have the same direction. Our main result is to show that any planarity preserving pseudo-morph consisting of unidirectional steps and contraction of low degree vertices can be turned into a true morph without increasing the number of steps. Using this, we strengthen Alamdari et al.'s result to use only unidirectional morphs, and in the process we simplify the proof.

Drawing partially embedded and simultaneously planar graphs

Abstract: We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph PEG problem--to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph--and the simultaneous planarity SEFE problem--to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding information about the drawing is given. Our result on partially embedded graph drawing generalizes a classic result of Pach and Wenger showing that any planar graph can be drawn with fixed locations for its vertices and with a linear number of bends per edge.

Practical experience with Hanani-Tutte for testing c-planarity

Abstract: We propose an algorithm for c-planarity testing which is correct and efficient, but not, in general, complete, i.e., there are input instances on which the algorithm declines to give an answer. At the core of this algorithm is an algebraic criterion based on work by the third author [20] with the following properties: (1) The criterion is a necessary condition for c-planarity, (2) for special graph classes, including c-connected graphs, the condition is also sufficient, and (3) the criterion can be tested efficiently in polynomial time. The algebraic criterion is not sufficient in general; however, we can extend it to a (still efficient) algorithm that verifies the answer of the criterion by building a c-planar embedding of the input graph. Our practical experiments show that this algorithm works well in practice. This is the first time that all instances from state-of-the-art benchmark sets for testing c-planarity are solved correctly. The algorithm is conceptually very simple and easy to implement.

Planarity of permutability graphs of subgroups of groups

Abstract: Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are planar. In addition, we classify the finite groups whose permutability graphs of subgroups are one of outerplanar, path, cycle, unicyclic, claw-free or C4-free. Also, we investigate the planarity of permutability graphs of subgroups of infinite groups.

Column Planarity and Partial Simultaneous Geometric Embedding

Abstract: We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any e>0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at mosti¾?5/6+en. We also consider a relaxation of simultaneous geometric embedding SGE, which we call partial SGE PSGE. A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE.

Power spectrum and non-Gaussianities in anisotropic inflation

Abstract: We study the planar regime of curvature perturbations for single field inflationary models in an axially symmetric Bianchi I background. In a theory with standard scalar field action, the power spectrum for such modes has a pole as the planarity parameter goes to zero. We show that constraints from back reaction lead to a strong lower bound on the planarity parameter for high-momentum planar modes and use this bound to calculate the signal-to-noise ratio of the anisotropic power spectrum in the CMB, which in turn places an upper bound on the Hubble scale during inflation allowed in our model. We find that non-Gaussianities for these planar modes are enhanced for the flattened triangle and the squeezed triangle configurations, but show that the estimated values of the fNL parameters remain well below the experimental bounds from the CMB for generic planar modes (other, more promising signatures are also discussed). For a standard action, fNL from the squeezed configuration turns out to be larger compared to that from the flattened triangle configuration in the planar regime. However, in a theory with higher derivative operators, non-Gaussianities from the flattened triangle can become larger than the squeezed configuration in a certain limit of the planarity parameter.

Exercises in Graph Theory

Abstract: Introduction. 1. ABC of Graph Theory. 2. Trees. 3. Independence and Coverings. 4. Connectivity. 5. Matroids. 6. Planarity. 7. Graph Traversals. 8. Degree Sequences. 9. Graph Colorings. 10. Directed Graphs. 11. Hypergraphs. Answers to Chapter 1: ABC of Graph Theory. Answers to Chapter 2: Trees. Answers to Chapter 3: Independence and Coverings. Answers to Chapter 4: Connectivity. Answers to Chapter 5: Matroids. Answers to Chapter 6: Planarity. Answers to Chapter 7: Graph Traversals. Answers to Chapter 8: Degree Sequences. Answers to Chapter 9: Graph Colorings. Answers to Chapter 10: Directed Graphs. Answers to Chapter 11: Hypergraphs. Bibliography. Index. Notations.

Loudspeaker Array Processing For Personal Sound Zone Reproduction.

Abstract: Sound zone reproduction facilitates listeners wishing to consume personal audio content within the same acoustic enclosure by filtering loudspeaker signals to create constructive and destructive interference in different spatial regions. Published solutions to the sound zone problem are derived from areas such as sound field synthesis and beamforming. The first contribution of this thesis is a comparative study of multi-point approaches. A new metric of planarity is adopted to analyse the spatial distribution of energy in the target zone, and the well-established metrics of acoustic contrast and control effort are also used. Simulations and experimental results demonstrate the advantages and disadvantages of the approaches. Energy cancellation produces good acoustic contrast but allows very little control over the target sound field; synthesis-derived approaches precisely control the target sound field but produce less contrast. Motivated by the limitations of the existing optimization methods, the central contribution of this thesis is a proposed optimization cost function ‘planarity control’, which maximizes the acoustic contrast between the zones while controlling sound field planarity by projecting the target zone energy into a spatial domain. Planarity control is shown to achieve good contrast and high target zone planarity over a large frequency range. The method also has potential for reproducing stereophonic material in the context of sound zones. The remaining contributions consider two further practical concerns. First, judicious choice of the regularization parameter is shown to have a significant effect on the contrast, effort and robustness. Second, attention is given to the problem of optimally positioning the loudspeakers via a numerical framework and objective function. The simulation and experimental results presented in this thesis represent a significant addition to the literature and will influence the future choices of control methods, regularization and loudspeaker placement for personal audio. Future systems may incorporate 3D rendering and listener tracking.

Picking Planar Edges; or, Drawing a Graph with a Planar Subgraph

Abstract: Given a graph G and a subset F⊆EG of its edges, is there a drawing of G in which all edges of F are free of crossings? We show that this question can be solved in polynomial time using a Hanani-Tutte style approach. If we require the drawing of G to be straight-line, but allow up to one crossing along each edge in F, the problem turns out to be as hard as the existential theory of the real numbers.

Degree sequence index strategy.

Abstract: We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the k-independence and the k-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and K1,r-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester.

5,5′-Bis-(alkylpyridinyl)-2,2′-bithiophenes: synthesis, liquid crystalline behaviour and charge transport

Abstract: Liquid crystalline materials with 5,5′-bis-(alkylpyridinyl)-2,2′-bithiophene structures with different positions of the nitrogen atoms in the aromatic core were synthesized, characterized by differential scanning calorimetry, polarized optical microscopy and, in some cases, by synchrotron X-ray diffraction analysis and single crystal X-ray analysis. The molecular planarity and liquid crystalline (LC) behaviour of these materials are strongly influenced by the location of the nitrogens in the aromatic core, and LC behaviour is more complex than the mesomorphic behaviour of the known structurally related 5,5′-bis-(alkylphenyl)-2,2′-bithiophene LCs. Single crystal X-ray structural analysis of 5,5′-bis-(5-alkylpyridin-2-yl)-2,2′-bithiophenes 2 reveals that the aromatic core is nearly planar with s-cis conformation of the thiophene and pyridine rings and s-trans conformation of the thiophene rings of the central 2,2′-bithiophene unit. The length of the alkyl chains has a pronounced effect on the molecular planarity and the packing motifs. A preliminary study of the charge-transporting properties of 5,5′-bis-(5-alkylpyridin-2-yl)-2,2′-bithiophenes 2d and 2e with n-nonyl and n-decyl alkyl chains, respectively, by the time-of-flight technique shows that the hole mobility is temperature independent, electric field dependent and with a magnitude of ∼1.5 × 10−4 cm2 V−1 s−1 in the unidentified S2/Cr2 phase.

Advantage of the N-Alkylation Strategy for Retaining the Molecular Planarity for Oligothiophene/Imidazole/1,10-Phenanthroline-Based Heterocyclic Semiconducting and Fluorescent Compounds

Abstract: A family of planar oligothiophene/imidazole/1,10-phenanthroline (OTIP)-based heterocyclic, aromatic, semiconducting, and fluorescent compounds with N-substituted alkyl chains (allyl, n-butyl, n-octyl, n-dodecyl, and n-cetyl) have been designed and synthesized. They all have specific N-coordination sites, various donor-acceptor spacers, good molecular planarity, suitable solubility, and high thermal stability. In comparison with conventional double β-alkylation of the thiophene ring, our results reveal that the single imidazole N-alkylation strategy for OTIPs has the advantage of maintaining the planarity of the whole molecule, in addition to improving the solubility, which can be clearly verified by the small dihedral angles between adjacent thiophene/imidazole/1,10-phenanthroline (TIP) rings in eight X-ray single-crystal structures. In particular, n-dodecyl- and n-cetyl-substituted OTIPs (7 and 8) with the same molecular length of 2.37 nm (MW =939 and 1052), show good molecular planarity with the aforementioned dihedral angles of 8.9(5) and 10.4(5)°. Furthermore, special attention has been paid to the physicochemical properties of seven symmetrical OTIPs (6-8, 13-15, and 19), including two to six thiophene rings in the middle of their molecular structures. To the best of our knowledge, this is the first synthetic, structural, and spectral investigation into the N-alkylation of OTIP-based compounds.

On the Complexity of HV-rectilinear Planarity Testing

Abstract: An HV-restricted planar graph G is a planar graph with vertex-degree at most four and such that each edge is labeled either H horizontal or V vertical. The HV-rectilinear planarity testing problem asks whether G admits a planar drawing where every edge labeled V is drawn as a vertical segment and every edge labeled H is drawn as a horizontal segment. We prove that HV-rectilinear planarity testing is NP-complete even for graphs having vertex degree at most three, which solves an open problem posed by both Manuch et al. GD 2010 and Durucher et al. LATIN 2014. We also show that HV-rectilinear planarity can be tested in polynomial time for partial 2-trees of maximum degree four, which extends a previous result by Durucher et al. LATIN 2014 about HV-restricted planarity testing of biconnected outerplanar graphs of maximum degree three. When the test is positive, our algorithm returns an orthogonal representation of G that satisfies the given H- and V-labels on the edges.

Morphing planar triangulations

Abstract: A morph between two drawings of the same graph can be thought of as a continuous deformation between the two given drawings. A morph is linear if every vertex moves along a straight line segment from its initial position to its final position. In this thesis we study algorithms for morphing, in which the morphs are given by sequences of linear morphing steps. In 1944, Cairns proved that it is possible to morph between any two planar drawings of a planar triangulation while preserving planarity during the morph [22]. However this morph may require exponentially many steps. It was not until 2013 that Alamdari et al. proved that the morphing problem for planar triangulations can be solved using polynomially many steps [2]. In 1990 it was shown by Schnyder [50, 51] that using special drawings that we call Schnyder drawings it is possible to draw a planar graph on a O(n) × O(n) grid, and moreover such drawings can be found in O(n) time (here n denotes the number of vertices of the graph). It still remains unknown whether there is an efficient algorithm for morphing in which all drawings are on a polynomially sized grid. In this thesis we give two different new solutions to the morphing problem for planar triangulations. Our first solution gives a strengthening of the result of Alamdari et al. where each step is a unidirectional morph. This also leads to a simpler proof of their result. Our second morphing algorithm finds a planar morph consisting of O(n) steps between any two Schnyder drawings while remaining in an O(n) × O(n) grid. However, there are drawings of planar triangulations which are not Schnyder drawings, and for these drawings we show that a unidirectional morph consisting of O(n) steps that ends at a Schnyder drawing can be found. We conclude this work by showing that the basic steps from our morphs can be implemented using a Schnyder wood and weight shifts on the set of interior faces.

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

Abstract: We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.

Upward-rightward planar drawings

Abstract: Upward drawing is a widely studied drawing convention for the visual representation of directed graphs In an upward drawing vertices are mapped to distinct points of the plane, and edges are curves monotonically increasing in the vertical direction, according to their orientation In particular, not all planar digraphs admit an upward planar drawing (ie, an upward drawing with no edge crossing), and testing whether a planar digraph is upward planar drawable is NP-hard Furthermore, straight-line upward planar drawings may require exponential area In this paper we study a relaxation of upward drawings, called upward-rightward drawings; in such a drawing for any directed path from a vertex u to a vertex v it must be that either v is above u or v is to the right of u In contrast with upward planarity, we prove that every planar digraph admits an upward-rightward planar drawing with straight-line edges and that this drawing can be computed in linear time and polynomial area

Morphing Schnyder Drawings of Planar Triangulations

Abstract: We consider the problem of morphing between two planar drawings of the same triangulated graph, maintaining straight-line planarity. A paper in SODA 2013 gave a morph that consists of On 2 steps where each step is a linear morph that moves each of the n vertices in a straight line at uniform speed [1]. However, their method imitates edge contractions so the grid size of the intermediate drawings is not bounded and the morphs are not good for visualization purposes. Using Schnyder embeddings, we are able to morph in On 2 linear morphing steps and improve the grid size to On×On for a significant class of drawings of triangulations, namely the class of weighted Schnyder drawings. The morphs are visually attractive. Our method involves implementing the basic "flip" operations of Schnyder woods as linear morphs.