Showing papers on "Planarity testing published in 2019"

A Survey on Graph Drawing Beyond Planarity

Abstract: Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of nonplanar graphs in terms of forbidden crossing configurations. The aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

First-principles theoretical designing of planar non-fullerene small molecular acceptors for organic solar cells: manipulation of noncovalent interactions.

Abstract: Non-fullerene small molecular acceptors (NFSMAs) exhibit promising photovoltaic performance; however, their electron mobilities are still relatively lower than those of fullerene derivatives. The construction of a highly planar conjugated system is an important strategy to achieve high charge mobility. In chemical parlance, it is tedious and costly to synthesize planar compounds by restricting the rotation at a specific bond. Recently, nonbonding intramolecular interactions, also termed "conformational locks," have been considered as an alternative way to achieve planar geometry. The successful implementation of this approach for designing polymers has been extensively reported. Recently, several examples of NFSMAs containing conformational locks have been presented in the literature. This situation encourages us to perform a detailed theoretical investigation in designing planar small molecular acceptors. Various nonbonding interactions were studied using accurate computational methods, and molecules with multiple nonbonding interactions showed high planarity. Planar acceptors showed red-shifted absorption with high oscillator strengths. In addition, backbone planarity plays a very important role in tuning the charge transport properties and decreasing reorganization energy. Our results could provide important information to guide the further design of promising NFSMA materials.

The Consequences of Twisting Nanocarbons: Lessons from Tethered Twisted Acenes

Abstract: ConspectusThe properties of polycyclic aromatic hydrocarbons are determined by their size, shape, and functional groups. Equally important is their curvature, since deviation from planarity can aff...

Track Layouts, Layered Path Decompositions, and Leveled Planarity

Abstract: We investigate two types of graph layouts, track layouts and layered path decompositions, and the relations between their associated parameters track-number and layered pathwidth. We use these two types of layouts to characterize leveled planar graphs, which are the graphs with planar leveled drawings with no dummy vertices. It follows from the known NP-completeness of leveled planarity that track-number and layered pathwidth are also NP-complete, even for the smallest constant parameter values that make these parameters nontrivial. We prove that the graphs with bounded layered pathwidth include outerplanar graphs, Halin graphs, and squaregraphs, but that (despite having bounded track-number) series–parallel graphs do not have bounded layered pathwidth. Finally, we investigate the parameterized complexity of these layouts, showing that past methods used for book layouts do not work to parameterize the problem by treewidth or almost-tree number but that the problem is (non-uniformly) fixed-parameter tractable for tree-depth.

Random walks and forbidden minors II: a poly(d ε-1)-query tester for minor-closed properties of bounded degree graphs

Abstract: Let G be a graph with n vertices and maximum degree d. Fix some minor-closed property P (such as planarity). We say that G is e-far from P if one has to remove e dn edges to make it have P. The problem of property testing P was introduced in the seminal work of Benjamini-Schramm-Shapira (STOC 2008) that gave a tester with query complexity triply exponential in e−1. Levi-Ron (TALG 2015) have given the best tester to date, with a quasipolynomial (in e−1) query complexity. It is an open problem to get property testers whose query complexity is (de−1), even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity d· (e−1). The previous line of work on (independent of n, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (FOCS 2018) analyzing random walk algorithms that find forbidden minors.

Emergent planarity in two-dimensional Ising models with finite-range Interactions

Abstract: The known Pfaffian structure of the boundary spin correlations, and more generally order–disorder correlation functions, is given a new explanation through simple topological considerations within the model’s random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on $${\mathbb {Z}}^2$$ with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

Shape Memory Activated Self-Deployable Solar Sails: Small-Scale Prototypes Manufacturing and Planarity Analysis by 3D Laser Scanner

Abstract: Solar sails are propellantless systems where the propulsive force is given by the momentum exchange of reflecting photons. Thanks to the use of shape memory alloys for the self-actuation of the system, complexity of the structure itself has decreased and so has the weight of the whole structure. Four self-deploying systems based on the NiTi shape memory wires have been designed and manufactured in different configurations (wires disposal and folding number). The deployed solar sails surfaces have been acquired by a Nextengine 3D Laser Scanner based on the Multistripe Triangulation. 3D maps have been pre-processed through Geomagic Studio and then elaborated in the Wolfram Mathematica environment. The planarity degree has been evaluated as level curves from the regression plane highlighting marked differences between the four configurations and locating the vertices as the most critical zones. These results are useful in the optimization of the best folding solution both in the weight/surface reduction and in the planarity degree of the solar sail.

Exact distance oracles for planar graphs with failing vertices

Abstract: We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X. We propose oracles that can handle any number k of failures. More specifically, for a directed weighted planar graph with n vertices, any constant k, and for any [MATH HERE], we propose an oracle of size [MATH HERE] that answers queries in O(q) time.1 In particular, we show an O(n)-size, [MATH HERE]-query-time oracle for any constant k. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures (k = 1), our [MATH HERE]-size, O(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for q = Ω(nt), t ∈ (1/4,1/2]. For multiple failures, no planarity exploiting results were previously known.

Generalized Planar Feynman Diagrams: Collections

Abstract: Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each degeneration of a tree leads to a single planar neighbor. Amplitudes are computed as an integral over the space of metrics where edge lengths are Schwinger parameters. In this work we propose that a natural generalization of Feynman diagrams is provided by what are known as metric tree arrangements. These are collections of metric trees subject to a compatibility condition on the metrics. We introduce the notion of planar collections of Feynman diagrams and argue that using planarity one can generate all planar collections starting from any one. Moreover, we identify a canonical initial collection for all $n$. Generalized $k=3$ biadjoint amplitudes, introduced by Early, Guevara, Mizera, and one of the authors, are easily computed as an integral over the space of metrics of planar collections of Feynman diagrams.

NodeTrix Planarity Testing with Small Clusters

Abstract: We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant k. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be arbitrarily chosen. We show that NodeTrix planarity testing with fixed sides can be solved in \(O(k^{3k+\frac{3}{2}} n^3)\) time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in \(O(n^3)\) time for \(k = 2\), but it is NP-complete for any \(k \ge 3\). NodeTrix planarity testing remains NP-complete also in the free side model when \(k > 4\).

Atomic Embeddability, Clustered Planarity, and Thickenability

Abstract: We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently, despite relentless efforts. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time. Our algorithm for atomic embeddability combines ideas from Carmesin's work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory. Finally, we give a polynomial-time reduction from atomic embeddability to thickenability thereby showing that both problems are polynomially equivalent, and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.

Extending Upward Planar Graph Drawings.

Abstract: In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing $\Gamma_H$ of a subgraph $H$ of a directed graph $G$ and asks whether $\Gamma_H$ can be extended to an upward planar drawing of $G$. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We show the following results. First, we prove that the Upward Planarity Extension problem is NP-complete, even if $G$ has a prescribed upward embedding, the vertex set of $H$ coincides with the one of $G$, and $H$ contains no edge. Second, we show that the Upward Planarity Extension problem can be solved in $O(n \log n)$ time if $G$ is an $n$-vertex upward planar $st$-graph. This result improves upon a known $O(n^2)$-time algorithm, which however applies to all $n$-vertex single-source upward planar graphs. Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which $G$ is a directed path or cycle with a prescribed upward embedding, $H$ contains no edges, and no two vertices share the same $y$-coordinate in $\Gamma_H$.

Influence of organic cation planarity on structural templating in hybrid metal-halides

Abstract: Controlling the connectivity and topology of solids is a versatile way to target desired physical properties. This is especially relevant in the realm of hybrid halide semiconductors, where the long-range connectivity of the inorganic substructural unit can lead to significant changes in optoelectronic properties such as photoluminescence, charge transport, and absorption. We present a new series of hybrid metal-halide semiconductors, (phenH2)BiI5·H2O, (2,2-bpyH2)BiI5, (BrbpyH)BiI4·H2O, (phenH2)2Pb3I10·2H2O, and (2,2-bpyH2)2Pb3I10 where (phenH2)2+ = 1,10-phenanthroline-1,10-diium, (2,2-bpyH2)2+ = 2,2'-bipyridine-1,1'-diium and (BrbpyH)+ = 6,6'-dibromo-2,2'-bipyridium. These compounds allow us to observe how the planarity of the cation, induced either through structural modification in the case of (phenH2)2+ or through non-covalent interactions in (BrbpyH)+, both relative to (2,2-bpyH2)2+, modifies the inorganic substructural unit. While the Pb2+ series of compounds show minimal changes in inorganic connectivity, we observe large differences in the Bi3+ series, ranging from 0-D dimers to corner- and edge-sharing 1-D chains of octahedra. We find that compounds containing (phenH2)2+ and (BrbpyH)+ pack more efficiently than those with (2,2-bpyH2)2+ due to their retention of planarity leading to greater inorganic connectivity. Electronic structure calculations and optical diffuse reflectance reveal that the band gaps of these compounds are influenced by the degree of inorganic connectivity and the inorganic substructural unit distances. These results show that the structure and planarity of organic cations can directly influence both the inorganic connectivity and the optical properties that could be tuned for certain optoelectronic applications.

Characteristic properties of ellipsoids and convex quadrics

Abstract: This is a survey on various characteristic properties of ellipsoids and convex quadrics in the family of convex hypersurfaces in $${\mathbb {R}}^n$$ . The topics under consideration include planar sections and projections, planarity conditions on midsurfaces and shadow-boundaries, intersections of homothetic copies, projective centers, and invariant mappings.

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. In “How to morph planar graph drawings” (SIAM Journal on Computing) the authors give a morph that consists of O(n) steps where each step is a linear morph that moves each of the n vertices in a straight line at uniform speed. 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 $$O(n^2)$$ linear morphing steps and improve the grid size to $$O(n)\times O(n)$$ 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.

Clustered Planarity with Pipes

Abstract: We study the version of the C-Planarity problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the Strip Planarity problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm.

Modification to Planarity is Fixed Parameter Tractable

Abstract: A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G| 2) steps. We also present several applications of our approach to related problems.

A theoretical approach to the role of different types of electrons in planar elongated boron clusters.

Abstract: We analyze the thermodynamic stability of some small elongated boron clusters and confirm the relationship between their planarity and their inherent electron configuration […σ2(n+1) π12(n+1) π22n]. Delocalized σ electrons in an elongated bare boron cluster and 2c-2e C–H bonds in a corresponding elongated hydrocarbon play a vital role in maintaining their planar structure. Through the eigenstates derived from a model of a particle moving in a rectangle, the rectangle model, our study suggests that the larger planar elongated boron clusters are not thermodynamically stable. A partition of the electron densities which is consistent with the electron count, points out that the dianionic, neutral and dicationic B102−/0/2+ clusters are doubly σ and π aromatic, singly π aromatic, and doubly σ and π antiaromatic, respectively.

Non-existence of bi-infinite geodesics in the exponential corner growth model

Abstract: This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

Effect of Planarity of Aromatic Rings Appended to o-Carborane on Photophysical Properties: A Series of o-Carboranyl Compounds Based on 2-Phenylpyridine- and 2-(Benzo[b]thiophen-2-yl)pyridine

Abstract: Herein, we investigated the effect of ring planarity by fully characterizing four pyridine-based o-carboranyl compounds. o-Carborane was introduced to the C4 position of the pyridine rings of 2-phenylpyridine and 2-(benzo[b]thiophen-2-yl)pyridine (CB1 and CB2, respectively), and the compounds were subsequently borylated to obtain the corresponding C∧N-chelated compounds CB1B and CB2B. Single-crystal X-ray diffraction analysis of the molecular structures of CB2 and CB2B confirmed that o-carborane is appended to the aryl moiety. In photoluminescence experiments, CB2, but not CB1, showed an intense emission, assignable to intramolecular charge transfer (ICT) transition between the aryl and o-carborane moieties, in both solution and film states. On the other hand, in both solution and film states, CB1B and CB2B demonstrated a strong emission, originating from π-π * transition in the aryl groups, that tailed off to 650 nm owing to the ICT transition. All intramolecular electronic transitions in these o-carboranyl compounds were verified by theoretical calculations. These results distinctly suggest that the planarity of the aryl groups have a decisive effect on the efficiency of the radiative decay due to the ICT transition.

Extending Upward Planar Graph Drawings

Abstract: In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing \(\varGamma _H\) of a subgraph H of a directed graph G and asks whether \(\varGamma _H\) can be extended to an upward planar drawing of G.

The planarity of heteroatom analogues of benzene: Energy component analysis and the planarization of hexasilabenzene.

Abstract: There are various nonplanar heteroatom analogues of benzene-cyclic 6π electron systems-and among them, hexasilabenzene (Si6 H6 ) is well known as a typical example. To determine the factors that control their planarity, quantum chemical calculations and an energy component analysis were performed. The results show that the energy components mainly controlling the planarity of benzene and hexasilabenzene are different. For hexasilabenzene, electron repulsion energy was found to be significantly important for the planarity. The application of the pseudo Jahn-Teller effect and the Carter-Goddard-Malrieu-Trinquier model for the interpretation of the planarity of the benzene analogues was also investigated. Furthermore, based on the quantitative results, it was revealed that the planarization of hexasilabenzene is realized by introducing substituents with π-accepting ability, such as the boryl group, that bring about a reduction of the π-electron repulsion on the silicon skeleton. © 2018 Wiley Periodicals, Inc.

A planarity estimate for pinched solutions of mean curvature flow

Abstract: We show that the blow-ups of compact solutions to the mean curvature flow in $\mathbb{R}^N$ initially satisfying the pinching condition $|A|^2 < c |H|^2$ for a suitable constant $c = c(n)$ must be codimension one.

Targeted and selective HOMO energy control by fine regulation of molecular planarity and its effect on the interfacial charge transfer process in dye-sensitized solar cells

Abstract: In terms of the in-depth development of organic dyes, targeted and selective energy control is becoming a more and more important objective. Herein, four indoline sensitizers based on D-π-A-π-A construction were designed and synthesized with exactly the same donor and acceptor segments. Their molecular planarity was regulated by introducing various side chains into donor bridges. Interestingly, along with an improvement of planarity at a donor bridge, the HOMO levels of the dyes lift gradually, and more importantly, their LUMO levels remain at around the same value. Besides, better molecular planarity is obviously preferred to obtain higher charge injection efficiency but, an overly planar molecule may cause an overly high HOMO level, leading to poor dye regeneration efficiency. Furthermore, an appropriate side chain also restrains charge recombination to some extent, while an overly large side chain gives more chance for I3- to recombine with charge in the conduction band. Accordingly, our results demonstrated that regulation of planarity at a donor bridge not only provides targeted and selective control of the HOMO of the dye, but also enable fine adjustment with multiple interfacial charge transfer processes. Molecular planarity deserves to play an important role in the design of organic dyes, providing a significant strategy for the further development of organic sensitizers.