Showing papers on "Planarity testing published in 2015"

Controlling Crystallite Orientation of Diketopyrrolopyrrole‐Based Small Molecules in Thin Films for Highly Reproducible Multilevel Memory Device: Role of Furan Substitution

Abstract: For the organic memory device with vertically arranged electrodes, controlling the film-packing to achieve highly oriented crystallite arrangement is critical but challenging for obtaining the satisfied performance. Here, the effect of backbone planarity on the crystallite orientation is studied. Two diketopyrrolopyrrole-based small molecules (NI2PDPP and NI2FDPP) are synthesized with increasing planarity by furan substitution for phenyl rings. Upon thin-film analysis by atomic force microscopy, X-ray diffraction, and grazing-incidence small-angle X-ray scattering, the orientations of these crystallites are demonstrated to be well controlled through tailoring molecular planarity. The highly planar NI2FDPP in film prefers out-of-plane crystallite orientation with respect to the substrate normal while the nonplanar NI2PDPP displays less ordered packing with a broad orientation distribution relative to the substrate. As a result, NI2FDPP-based memory device exhibits superior multilevel performance. More importantly, the oriented crystallite arrangement favors uniformity in NI2FDPP thin film, thus, the device displays higher reproducibility of memory effects. This study provides an effective synthetic strategy for designing multilevel memory materials with favorable crystallite orientation.

A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

Abstract: Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient partition oracles. A partition oracle is a procedure that, given access to the incidence lists representation of a bounded-degree graph G= (V,E) and a parameter e, when queried on a vertex v ∈ V, returns the part (subset of vertices) that v belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most e |V|. In this work, we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/e, improving on the result of Hassidim et al. (Proceedings of FOCS 2009), who gave a partition oracle with query complexity exponential in 1/e. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.

Testing Planarity of Partially Embedded Graphs

Abstract: We study the following problem: given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph Gq This problem fits the paradigm of extending a partial solution for a problem to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes an otherwise easy problem hard, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmas, which show that the planarity of partially embedded graphs exhibits the ‘TONCAS’ behavior “the obvious necessary conditions for planarity are also sufficient.” These conditions are expressed in terms of the interplay between (1) the rotation system and containment relationships between cycles and (2) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we make our algorithm run in linear time.Finally, we consider several generalizations of the problem, such as minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. We also apply our algorithm to the simultaneous graph drawing problem Simultaneous Embedding with Fixed Edges (Sefe). There we obtain a linear-time algorithm for the case that one of the input graphs or the common graph has a fixed planar embedding.

Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems

Abstract: In this article, we define and study the new problem of Simultaneous PQ-Ordering. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations between these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. Simultaneous PQ-Ordering asks whether orders of the leaves of each of the trees can be chosen simultaneously; that is, for every child-parent relation, the order chosen for the parent is an extension of the order chosen for the child. We show that Simultaneous PQ-Ordering is NP-complete in general, and we identify a family of instances that can be solved efficiently, the 2-fixed instances. We show that this result serves as a framework for several other problems that can be formulated as instances of Simultaneous PQ-Ordering. In particular, we give linear-time algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover, we obtain a linear-time algorithm for Partially PQ-Constrained Planarity for biconnected graphs, which asks for a planar embedding in the presence of PQ-trees that restrict the possible orderings of edges around vertices, and a quadratic-time algorithm for Simultaneous Embedding with Fixed Edges for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two nontrivial blocks. This includes, for example, the case where both graphs have a maximum degree of 5.

Tight Bounds for Graph Problems in Insertion Streams

Abstract: Despite the large amount of work on solving graph problems in the data stream model, there do not exist tight space bounds for almost any of them, even in a stream with only edge insertions. For example, for testing connectivity, the upper bound is O(n * log(n)) bits, while the lower bound is only Omega(n) bits. We remedy this situation by providing the first tight Omega(n * log(n)) space lower bounds for randomized algorithms which succeed with constant probability in a stream of edge insertions for a number of graph problems. Our lower bounds apply to testing bipartiteness, connectivity, cycle-freeness, whether a graph is Eulerian, planarity, H-minor freeness, finding a minimum spanning tree of a connected graph, and testing if the diameter of a sparse graph is constant. We also give the first Omega(n * k * log(n)) space lower bounds for deterministic algorithms for k-edge connectivity and k-vertex connectivity; these are optimal in light of known deterministic upper bounds (for k-vertex connectivity we also need to allow edge duplications, which known upper bounds allow). Finally, we give an Omega(n * log^2(n)) lower bound for randomized algorithms approximating the minimum cut up to a constant factor with constant probability in a graph with integer weights between 1 and n, presented as a stream of insertions and deletions to its edges. This lower bound also holds for cut sparsifiers, and gives the first separation of maintaining a sparsifier in the data stream model versus the offline model.

Steric minimization towards high planarity and molecular weight for aggregation and photovoltaic studies

Abstract: Taking into account the effect of planarity, alkyl chain steric hindrance and molecular weight towards high performance polymer solar cells, a planar semiconducting polymer PTOBDTDTffBT is designed. Minimal intra- and inter-molecular steric hindrances are realized by removing the alkyl chains at the 4,4'-positions of the DTffBT monomer and introducing shorter octyloxy chains on the BDT unit, respectively. Such a steric minimization strategy endows PTOBDTDTffBT with a high number average molecular weight of 343.37 kg mol(-1), a planar conjugated backbone and strong inter-molecular aggregation characteristics as well. The optical properties indicate that the aggregation can only be partially broken even at 160 C; theoretical calculations indicate that the polymer backbone has small distortion and the thiophene-phenyl bridge can move the octyloxy chains outward by 6.77 A from the polymer backbone, both of which ensure the strong inter-chain aggregation. In spite of the high planarity and strong aggregation of the polymer, PTOBDTDTffBT is relatively amorphous in its film form, indicated by grazing incidence X-ray diffraction analysis. PTOBDTDTffBT exhibits a narrow bandgap of 1.71 eV together with a high ionization potential of 5.46 eV. Because of the strong aggregation of the polymer, the PTOBDTDTffBT/PC71BM active layer exhibits temperature-dependent photovoltaic performance. When the active layer is spin-coated at a relatively low temperature below 100 degrees C, PTOBDTDTffBT/PC71BM exhibits a stable power conversion efficiency above 6.6%. However, when the temperature is elevated from 120 degrees C to 160 degrees C, the power conversion efficiency decreased gradually to 3.58% because of the decreased short-circuit current density and fill factor. The temperature-dependent photovoltaic performance is explained by the fact that the PC71BM is easily spun out from the solution at higher temperatures as confirmed by the UV-vis absorption and X-ray photoelectron spectroscopy analysis. High temperatures can also lead to coarse morphology and large phase separation in the active layer observed from the atomic force microscopy and transmission electron microscopy images. Finally, a maximum power conversion efficiency of 7.68% is obtained when the active layer is spin-coated at 80 degrees C, which is a significant improvement of the power conversion efficiency for the planar PBDTDTffBT series.

There are Plane Spanners of Degree 4 and Moderate Stretch Factor

Abstract: Let $${\fancyscript{E}}$$E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant $$t \ge 1$$t?1, a spanning subgraph $$G$$G of $${\fancyscript{E}}$$E is said to be a $$t$$t-spanner, or simply a spanner, if for any pair of nodes $$u,v$$u,v in $${\fancyscript{E}}$$E the distance between $$u$$u and $$v$$v in $$G$$G is at most $$t$$t times their distance in $${\fancyscript{E}}$$E. The constant $$t$$t is referred to as the stretch factor. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of $${\fancyscript{E}}$$E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper, we show that the complete Euclidean graph always contains a plane spanner of maximum degree 4 and make a big step toward closing the question. The stretch factor of the spanner is bounded by $$156.82$$156.82. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's $$L_1$$L1-Delaunay triangulation.

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 result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani–Tutte theorem in the case when each cluster induces a connected subgraph. Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident with at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.

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.

Scanning laser planarity detection

Abstract: A scanning laser projector (100) includes a proximity sensor and a planarity detector. When the proximity sensor detects an object closer than a proximity threshold, laser power is turned down. The scanning laser projector (100) can measure distance at a plurality of projection points in the projector's field of view. If the projection points lie substantially in a plane, laser power may be turned back up.

A Holant Dichotomy: Is the FKT Algorithm Universal?

Abstract: We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a holographic transformation a \emph{universal} strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems, Holant, and #CSP. A recurring theme has been that a holographic reduction to FKT is a universal strategy. Surprisingly, for planar Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In previous work, an important tool was a dichotomy for #CSP^d, which denotes #CSP where every variable appears a multiple of d times. However its proof violates planarity. We prove a dichotomy for planar #CSP^2. We apply this planar #CSP^2 dichotomy in the proof of the planar Holant dichotomy. As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is polynomial-time computable when the incidence graph is planar and k >= 5. The same problem is #P-hard when k=3 or k=4, which is also a consequence of our dichotomy. When k=2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor of all hyperedge sizes is at least 5.

A correspondence between rooted planar maps and normal planar lambda terms

Abstract: A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no beta-redexes, and planar if it is linear and the use of variables moreover follows a deterministic stack discipline. We begin by showing that the sequence counting normal planar lambda terms by a natural notion of size coincides with the sequence (originally computed by Tutte) counting rooted planar maps by number of edges. Next, we explain how to apply the machinery of string diagrams to derive a graphical language for normal planar lambda terms, extracted from the semantics of linear lambda calculus in symmetric monoidal closed categories equipped with a linear reflexive object or a linear reflexive pair. Finally, our main result is a size-preserving bijection between rooted planar maps and normal planar lambda terms, which we establish by explaining how Tutte decomposition of rooted planar maps (into vertex maps, maps with an isthmic root, and maps with a non-isthmic root) may be naturally replayed in linear lambda calculus, as certain surgeries on the string diagrams of normal planar lambda terms.

Beyond Level Planarity

Abstract: In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of $\textit{Cyclic Level Planarity}$ and $\textit{Torus Level Planarity}$, are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the $\textit{Simultaneous Level Planarity}$ of a set of level graphs, with respect to both the number of level graphs and the number of levels.

Non-planarity and metric Diophantine approximation for systems of linear forms

Abstract: In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of mxn matrices over R is introduced and studied. This notion generalises the one of non-planarity in R^n and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

Abstract: The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.

Upward Planarity Testing in Practice: SAT Formulations and Comparative Study

Abstract: A directed acyclic graph (DAG) is upward planar if it can be drawn without any crossings while all edges—when following them in their direction—are drawn with strictly monotonously increasing y-coordinates. Testing whether a graph allows such a drawing is known to be NP-complete, and while the problem is polynomial-time solvable for special graph classes, there is not much known about solving the problem for general graphs in practice. The only attempt so far has been a branch-and-bound algorithm over the graph’s triconnectivity structure, which was able to solve small graphs. Furthermore, there are some known FPT algorithms to deal with the problem.In this article, we propose two fundamentally different approaches based on the seemingly novel concept of ordered embeddings and on the concept of a Hanani-Tutte-type characterization of monotone drawings. In both approaches, we model the problem as special SAT instances, that is, logic formulae for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs.For the first time, we give an extensive experimental comparison between virtually all known approaches to the problem. To this end, we also investigate implementation issues and different variants of the known algorithms as well as of our SAT approaches and evaluate all algorithms on real-world as well as on constructed instances. We also give a detailed performance study of the novel SAT approaches. We show that the SAT formulations outperform all known approaches for graphs with up to 400 edges. For even larger graphs, a modified branch-and-bound algorithm becomes competitive.

On interpretation of protein X-ray structures: Planarity of the peptide unit.

Abstract: Pauling's mastery of peptide stereochemistry—based on small molecule crystal structures and the theory of chemical bonding—led to his realization that the peptide unit is planar and then to the Pauling–Corey–Branson model of the α-helix. Similarly, contemporary protein structure refinement is based on experimentally determined diffraction data together with stereochemical restraints. However, even an X-ray structure at ultra-high resolution is still an under-determined model in which the linkage among refinement parameters is complex. Consequently, restrictions imposed on any given parameter can affect the entire structure. Here, we examine recent studies of high resolution protein X-ray structures, where substantial distortions of the peptide plane are found to be commonplace. Planarity is assessed by the ω-angle, a dihedral angle determined by the peptide bond (CN) and its flanking covalent neighbors; for an ideally planar trans peptide, ω = 180°. By using a freely available refinement package, Phenix [Afonine et al. (2012) Acta Cryst. D, 68:352–367], we demonstrate that tightening default restrictions on the ω-angle can significantly reduce apparent deviations from peptide unit planarity without consequent reduction in reported evaluation metrics (e.g., R-factors). To be clear, our result does not show that substantial non-planarity is absent, only that an equivalent alternative model is possible. Resolving this disparity will ultimately require improved understanding of the deformation energy. Meanwhile, we urge inclusion of ω-angle statistics in new structure reports in order to focus critical attention on the usual practice of assigning default values to ω-angle constraints during structure refinement. Proteins 2015; 83:1687–1692. © 2015 Wiley Periodicals, Inc.

Planarity of Streamed Graphs

Abstract: In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A streamed graph is a stream of edges $$e_1,e_2,\dots ,e_m$$ on a vertex set $$V$$. A streamed graph is $$\omega $$-stream planar with respect to a positive integer window size $$\omega $$ if there exists a sequence of planar topological drawings $$\varGamma _i$$ of the graphs $$G_i=V,\{e_j \mid i\le j < i+\omega \}$$ such that the common graph $$G^{i}_\cap =G_i\cap G_{i+1}$$ is drawn the same in $$\varGamma _i$$ and in $$\varGamma _{i+1}$$, for $$1\le i < m-\omega $$. The Stream Planarity Problem with window size $$\omega $$ asks whether a given streamed graph is $$\omega $$-stream planar. We also consider a generalization, where there is an additional backbone graph whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the Stream Planarity Problem is $$\mathcal {NP}$$-completeeven when the window size is a constant and that the variant with a backbone graph is $$\mathcal {NP}$$-completefor all $$\omega \ge 2$$. On the positive side, we provide $$On+\omega {}m$$-time algorithms for i the case $$\omega = 1$$ and ii all values of $$\omega $$ provided the backbone graph consists of one $$2$$-connected component plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the Hanani-Tutte-style $$Onm^3$$-time algorithm proposed by Schaeferi¾ź[GD'14] for $$\omega =1$$.

An algorithm and upper bounds for the weighted maximal planar graph problem

Abstract: In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.

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.

The Utility of Untangling

Abstract: In this note we show how techniques developed for untangling planar graphs by Bose et al. [Discrete & Computational Geometry 42(4): 570-585 (2009)] and Goaoc et al. [Discrete & Com- putational Geometry 42(4): 542-569 (2009)] imply new results about some recent graph drawing models. These include column planarity, universal point subsets, and partial simultaneous geometric embeddings (with or without mappings). Some of these results answer open problems posed in previous papers.

Column Planarity and Partial Simultaneous Geometric Embedding for Outerplanar Graphs

Abstract: Given a graph G = (V,E), a set R V is column planar in G if we can assign x-coordinates to the vertices in R such that every assignment of y-coordinates to R gives a partial embedding of G that can be completed to a plane straight-line embedding of the whole graph. This notion is strongly related to unlabeled level planarity. We prove that every outerplanar graph on n vertices contains a column planar set of size at least n/2. We use this result to show that every pair of outerplanar graphs G1 and G2 on the same set V of n vertices admit an (n/4)-partial simultaneous geometric embedding (PSGE): a plane straight-line embedding of G1 and a plane straight-line embedding of G2 such that n/4 vertices are mapped to the same point in the two drawings. This is a relaxation of the well-studied notion of simultaneous geometric embedding, which is equivalent to n-PSGE.

A co-ideal based identity-summand graph of a commutative semiring

Abstract: Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $\Gamma_{I} (R)$ be a graph with the set of vertices $S_{I} (R) = \{x \in R\setminus I: x + y \in I$ for some $y \in R \setminus I\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$. We look at the diameter and girth of this graph. Also we discuss when $\Gamma_{I} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

Tutte relations, TQFT, and planarity of cubic graphs

Abstract: It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte's relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte's upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley-Lieb algebra and more generally on methods of topological quantum field theory.