Topological Graph Theory Research Articles

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that | E ( G )| ≥ α | V ( G )| 2 for some fixed α > 0. While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given ε > 0, returns an integer g such that G has an embedding in a surface of genus g , and this is ε-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G) \) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G) \) . The running time of the algorithm is O ( f (ε) n 2 ), where f (·) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time O ( f 1 (ε) n 2 ). Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [25]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [17] and to Fox, Lovász, and Zhao [14, 15]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ε-approximation ν ( G ) for the maximum number of edge-disjoint triangles in G . The value ν ( G ) can be computed by solving a linear program whose size is bounded by certain value f 2 (ε) depending only on ε. After solving the linear program, the genus can be approximated (see Corollary). The proof of this result is long and will be of independent interest in topological graph theory.

A python based algorithmic approach to optimize sulfonamide drugs via mathematical modeling

This article explores the structural properties of eleven distinct chemical graphs that represent sulfonamide drugs using topological indices by developing python algorithm. To find significant relationships between the topological characteristics of these networks and the characteristics of the associated sulfonamide drugs. We use quantitative structure-property relationship (QSPR) approaches. In order to model and forecast these correlations and provide insights into the structure-activity relationships that are essential for drug design and optimization, linear regression is a vital tool. A thorough framework for comprehending the molecular characteristics and behavior of sulfonamide drugs is provided by the combination of topological indices, graph theory and statistical models which advances the field of pharmaceutical research and development.

The maximum genus of the generalized Petersen Graph, GP (<em>n, k</em>) for the cases <em>k</em> = 1, 2

In Topological graph theory, the maximum genus of graphs has been a fascinating subject. For a simple connected graph G, the maximum genus γM(G) is the largest genus of an orientable surface on which G has a 2-cell embedding. γM(G) has the upper bound, γM(G)≤[β/2], where β(G) denotes the Betti number and G is said to be upper embeddable if the equality holds. In this study, the maximum genus of GP(n, k) is established as γM(GP(n,k))=[(n+1)/2] for k = 1 and k = 2 by proving the upper embeddability of generalized Petersen graph, GP(n, k) for the cases k = 1 and k = 2. The proof is done by obtaining spanning trees T and examining the components in the edge complements GP(n, k)\T for the cases k = 1 and k = 2 of GP(n, k).

Multimatroids and Rational Curves with Cyclic Action

Abstract We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-$A$ permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type-$B$ permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of ${\mathbb {R}}$-multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids.

On planarity of graphs in homotopy type theory

AbstractIn this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

How to frame innovation in mathematics

We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. In mathematical proofs, we distinguish two kinds of frames, namely structural frames and ontological frames. The interaction between both kinds of frames can drive mathematical interpretation. We first discuss two examples where structural frames (formulaic notation) drive ontological development (the discovery or exploration of mathematical objects). The development of Boole’s Boolean algebra may at first appear as a metaphorical treatment of the (then) new area of logic. In the analysis, we discuss how different (aspects of) certain algebraic frames change in the transfer, how arising difficulties are solved and overall argue that Boole uses the numerical algebra frame as a research template for the discovery of a system for calculations in logic. Following Ifrah, we analyse the discovery of zero as an extension to the number ontology as driven by the development of notation. Both structural and ontological frames are extended and simplified as notation progresses. Finally, we discuss two examples from infinite combinatorics, viz. topological graph theory, and one foundational issue. In both examples, the two simultaneous frames about one object are maintained independently. They motivate different research questions, but may also fruitfully interact: shifting between multiple synchronously maintained perspectives acts as a motor of innovation. The analysis shows how a frame-based approach allows to model how different perspectives drive mathematical innovation because they highlight different aspects, questions and heuristics.

Design of a class of reconfigurable hybrid mechanisms for large complex curved surface machining based on topological graph theory

A design method for hybrid mechanism configuration synthesis based on topological graph theory is proposed to address the challenge of machining large and complex curved surfaces in the aerospace domain. The method leads to the generation of a new class of reconfigurable large extension hybrid mechanisms. Firstly, the spatial mechanism topological graph (SMTG) is obtained by evolving the structure of chemical molecules, which is used to express the spatial mechanism topological relationships. Then, combining graph theory methods with the SMTG, the topological relationships between motion modules of the hybrid mechanism are expressed, and the mechanisms of each motion module are designed. Finally, mechanisms of different functional modules are topologically connected and combined using the proposed topological connection relationship for the motion modules of hybrid mechanisms, thus resulting in a class of reconfigurable hybrid mechanisms (RHMs) with large extensibility. The RHMs exhibit strong structural stability, a large working space, and a small volume of occupation. It can be installed on the ends of industrial robots or in large guideways in the future to achieve high-precision machining of large complex curved structures.

Hausdorff of a cycle in topological graph

The main purpose of the paper is to bring together two areas in which strong relation, graph theory and topological spaces. And derive interesting formula for the set that contains all minimal dominating sets(MDS) and ( -set). Some separation axioms are discussed in topological graph theory, especially in a cycle graph Cn.

Eulerian and Bipartite Binary Delta-matroids

Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the concepts of Eulerian and bipartite delta-matroids and then extend the result from embedded graphs to arbitrary binary delta-matroids. The dual of any bipartite embedded graph is Eulerian. We also extend the result from embedded graphs to the class of delta-matroids that arise as twists of binary matroids. Several related results are also obtained.

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries

Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.

Atomic Embeddability, Clustered Planarity, and Thickenability

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. 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.

Partial-dual polynomials and signed intersection graphs

AbstractRecently, Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph as a generating function that enumerates all partial duals of the ribbon graph by Euler genus. It is analogous to the extensively studied polynomial in topological graph theory that enumerates by Euler genus all embeddings of a given graph. To investigate the partial-dual polynomial, one only needs to focus on bouquets: that is, ribbon graphs with exactly one vertex. In this paper, we shall further show that the partial-dual polynomial of a bouquet essentially depends on the signed intersection graph of the bouquet rather than on the bouquet itself. That is to say, two bouquets with the same signed intersection graph have the same partial-dual polynomial. We then give a characterisation of when a bouquet has a planar partial dual in terms of its signed intersection graph. Finally, we consider a conjecture posed by Gross, Mansour and Tucker that there is no orientable ribbon graph whose partial-dual polynomial has only one nonconstant term; this conjecture is false, and we give a characterisation of when all partial duals of a bouquet have the same Euler genus.

Design of all-wheel-drive power-split hybrid configuration schemes based on hierarchical topology graph theory

Power-split hybrid electric vehicles (PS-HEVs) can decouple the rotation speed and torque of the engine and power output shaft to ensure that the engine can work at efficient operating points as much as possible. All-wheel-drive (AWD) PS-HEVs can recover braking energy and have good trafficability, giving them good market application prospects. In this paper, we propose a powertrain configuration scheme design method based on hierarchical topological graph theory. Through kinematics and dynamics modeling, configuration mode classification, and scheme synthesis of PS-HEV powertrains, AWD PS-HEV configuration schemes are designed. The dynamic programming (DP) and rapid DP (Rapid-DP) algorithms are used to optimize the configuration schemes. Simulation results show that the AWD PS-HEVs designed by this method will have better fuel economy and acceleration performance than existing designs.

The average genus for bouquets of circles and dipoles

The bouquet of circles $B_n$ and dipole graph $D_n$ are two important classes of graphs in topological graph theory. For $n\geq 1$, we give an explicit formula for the average genus $\gamma_{avg}(B_n)$ of $B_n$. By this expression, one easily sees $\gamma_{avg}(B_n)=\frac{n-\ln n-c+1-\ln 2}{2}+o(1)$, where $c$ is the Euler constant. Similar results are obtained for $D_n$. Our method is new and deeply depends on the knowledge in ordinary differential equations.

An application of persistent homology and the graph theory to linguistics: The case of Tifinagh and Phoenician scripts

Abstract As the origin of the Tifinagh script remains uncertain, this work aims at exploring its probable relatedness with the Phoenician script. Using tools from within topological data analysis and graph theory, the similarity between the two scripts is studied. The clustering of their letter shapes is performed based on the pairwise distances between their topological signatures. The ideas presented in this work can be extended to study the similarity between any two writing systems and as such can serve as the first step for linguists to determine the possibly related scripts before conducting further analysis.

Rough continuity and rough separation axioms in $G_m$-closure approximation spaces

The theory of general topology view for continuous mappings is general version and is applied for topological graph theory. Separation axioms can be regard as tools for distinguishing objects in information systems. Rough theory is one of map the topology to uncertainty. The aim of this work is to presented graph, continuity, separation properties and rough set to put a new approaches for uncertainty. For the introduce of various levels of approximations, we introduce several levels of continuity and separation axioms on graphs in $G_m$-closure approximation spaces.

Designing Multi-Mode Power Split Hybrid Electric Vehicles Using the Hierarchical Topological Graph Theory

Power split hybrid electric vehicles (PS-HEVs) dominate the US and Japanese HEV market because of their superior fuel economy and drivability. In recent years, multi-mode PS-HEVs are offered by Toyota and GM. With multiple modes, it is possible for PS-HEVs to have both good launching/towing performance and fuel economy. Multiple modes are achieved by adding clutches or brakes. However, the corresponding design space can be quite large. To expedite the design process, a hierarchical topological graph theory approach is developed to systematically design a multi-mode PS-HEV with two planetary gear sets (PGSs). The process consists of three steps: 1) model the hybrid powertrain, 2) generate the multi-mode designs with specific modes, and 3) evaluate the performance of design candidates. In the performance screening process, designs are examined using the dynamic programming (DP) algorithm to evaluate their acceleration performance (0-100 km/h); and then a rapid dynamic programming (Rapid-DP) approach is used to compute their fuel economy under a specific driving cycle. Designs that pass the screening will then be retained as final vehicle designs. This design process ensures that the best designs are found and used.

Enumerating Abelian Typical Cube-Free Fold Coverings of a Circulant Graph

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. A covering projection p from a Cayley graph Cay(Γ, X) onto another Cayley graph Cay(Q, Y) is called typical if the function p : Γ → Q on the vertex sets is a group epimorphism. A typical covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. Recently, the equivalence classes of connected abelian typical prime-fold coverings of a circulant graph are enumerated. As a continuation of this work, we enumerate the equivalence classes of connected abelian typical cube-free fold coverings of a circulant graph.

Twenty Years of Progress of $$\hbox {JCDCG}^3$$

The conferences in the series of the Japan Conference on Discrete and Computational Geometry, Graphs and Games ($$\hbox {JCDCG}^3$$) have been held annually since 1997, except for 2008. Since 1997, almost one thousand research results have been presented in total at the conferences, and 11 post-conference proceedings and 6 special issues of journals have been published. To celebrate the 20th Anniversary of $$\hbox {JCDCG}^3$$, a summary of the notable results published in those proceedings are presented in this article. We focus on six areas such as games and puzzles, dissection and reversibility, foldings and unfoldings, point sets, visibility, and geometric and topological graph theory.

Network Coincidence Analysis: The netCoin R Package

The aim of the R package netCoin is to explore data structures using a number of statistical techniques that share the handling of interdependent variables. The main objective of this analysis is to detect events, characters, objects, attributes or characteristics that tend to appear together within a given set of scenarios. Its most notable feature is the combination of traditional multivariate statistical analysis and network analysis supported by topological graph theory. In addition, netCoin produces HTML graphs using the D3.js visualization library to support the interactive exploration of networked data. Among its many applications, netCoin can be used for the analysis of multiple responses in questionnaires to explore relevant associations, for the development of textual networks, for the study of ecological communities, for audience analysis, for mining large databases or for basket market analysis.