Journal articles: 'Faces In Planar Graph'

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Majeed, Amir Sabir. "Embedding of Neutrosophic Graphs on Topological Surfaces." JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences 32, no. 2 (2024): 183–96. http://dx.doi.org/10.29196/jubpas.v32i2.5279.

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Background: A planar graph (PG) is a graph with no intersecting edges. Particular to both crisp and neutrosophic graphs (NG) is the planar graph, in contrast to crisp planar graphs. NPGs allow for the intersection of neutrosophic edge NEs, since the value of planarity in these graphs is the degree of planarity of the intersected NEs. The NPGs are often represented on a flat surface. Materials and Methods: This study discusses how to embed NGs on surfaces such as spheres and m-toruses by defining the degree of intersection of the neutrosophic edges of NGs with finding the faces on the given graph structures using Euler's theorems. Here, the proofs of Euler's theorems help us find, given the total NFV of G, the interval containing that value. Result: As result of this work obtained that for any two isomorphic planer graphs, they have the same planarity value. For any neutrosophic planer graph with f = (1,1,1) can be embedded in the plane if it can be embedded in the sphere and according to NPGs, for planar and spherical surfaces, equivalent theorems to Euler's formula are proved and shown. Conclusion: It concludes that by using neutrosophic sets and crisp graphs to construct neutrosophic graphs with the benefit of Euler’s theorem, it can provide the concept of embedding neutrosophic graphs in different topological surfaces such as a plane, sphere, and m-torus.

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BIEDL, THERESE, and MARTIN VATSHELLE. "THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS." International Journal of Computational Geometry & Applications 23, no. 04n05 (2013): 357–95. http://dx.doi.org/10.1142/s0218195913600091.

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In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

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Ghorbani, Modjtaba, Matthias Dehmer, Shaghayegh Rahmani, and Mina Rajabi-Parsa. "A Survey on Symmetry Group of Polyhedral Graphs." Symmetry 12, no. 3 (2020): 370. http://dx.doi.org/10.3390/sym12030370.

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Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. We show that the order of symmetry group of such graphs divides 240.

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Wafiq, Hibi. "Girth Inequality In Planar Graphs." Multicultural Education 7, no. 9 (2021): 74. https://doi.org/10.5281/zenodo.5475229.

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As defined ingraph theory, the girth of a given graph is the length of its shortest cycle. Most of the papers, that mentioned the girth of a graph, I will mention here[1, 2, 3 and 4], investigated other features when the girth was one of the data, or described algorithms to find the girth when the graph is given. The purpose of this paper is to give inequalities, which explains girth’s relationship with the number of edges, the number of vertices and the number of faces, in a planar and connected graph.

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Taheri-Dehkordi, Meysam, and Gholam Hossein Fath-Tabar. "Nice pairs of pentagons in chamfered fullerenes." MATCH Communications in Mathematical and in Computer Chemistry 87, no. 3 (2021): 621–28. http://dx.doi.org/10.46793/match.87-3.621t.

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Fullerenes graphs are 3-connected, 3-regular planar graphs with faces including only pentagons and hexagons. If be a graph with a perfect matching, a subgraph H of G is a nice subgraph if G-V(H) has a perfect matching. In this paper, we show that in every fullerene graph arising from smaller fullerenes via chamfer transformation, each pair of pentagons is a nice subgraph.

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Muhiuddin, G., Saira Hameed, Ayman Rasheed, and Uzma Ahmad. "Cubic Planar Graph and Its Application to Road Network." Mathematical Problems in Engineering 2022 (July 11, 2022): 1–12. http://dx.doi.org/10.1155/2022/5251627.

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In this research article, we present the notion of a cubic planar graph and investigate its related properties. The cubic graphs are more effective than both interval-valued and fuzzy graphs as it represents the level of participation (membership degree) of vertices and edges both in interval form and as a fuzzy number. Moreover, it handles the uncertainty and vagueness more efficiently than both interval-valued fuzzy graph and fuzzy graph. The interval indicates a continuous process, whereas the point indicates a specific process. We introduce the terms cubic multigraph, cubic strong and weak edges, and degree of planarity for cubic planar graphs. Some fundamental theorems based on these concepts are also elaborated. We also propose the idea of a cubic strong and weak fuzzy faces and cubic dual graph. Some results related to these concepts are also established. Comparison with the existing method shows the worth of our proposed work.

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Cowan, Richard, and Simone Chen. "The random division of faces in a planar graph." Advances in Applied Probability 28, no. 2 (1996): 377–83. http://dx.doi.org/10.2307/1428062.

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A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

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Cowan, Richard, and Simone Chen. "The random division of faces in a planar graph." Advances in Applied Probability 28, no. 02 (1996): 377–83. http://dx.doi.org/10.1017/s0001867800048527.

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A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

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Casinillo, Emily L., and Leomarich F. Casinillo. "A Note on Full and Complete Binary Planar Graphs." Indonesian Journal of Mathematics Education 3, no. 2 (2020): 70. http://dx.doi.org/10.31002/ijome.v3i2.2800.

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Let G=(V(G), E(G)) be a connected graph where V(G) is a finite nonempty set called vertex-set of G, and E(G) is a set of unordered pairs {u, v} of distinct elements from V(G) called the edge-set of G. If is a connected acyclic graph or a connected graph with no cycles, then it is called a tree graph. A binary tree Tl with l levels is complete if all levels except possibly the last are completely full, and the last level has all its nodes to the left side. If we form a path on each level of a full and complete binary tree, then the graph is now called full and complete binary planar graph and it is denoted as Bn, where n is the level of the graph. This paper introduced a new planar graph which is derived from binary tree graphs. In addition, a combinatorial formula for counting its vertices, faces, and edges that depends on the level of the graph was developed.

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Cowan, Richard, and Simone Chen. "The random division of faces in a planar graph." Advances in Applied Probability 28, no. 2 (1996): 331. http://dx.doi.org/10.2307/1428040.

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Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

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Cowan, Richard, and Simone Chen. "The random division of faces in a planar graph." Advances in Applied Probability 28, no. 02 (1996): 331. http://dx.doi.org/10.1017/s0001867800048229.

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Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

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Sittitrai, Pongpat, Keaitsuda Maneeruk Nakprasit, and Kittikorn Nakprasit. "Partitioning Planar Graphs Without Specific Cycles into a Forest and a Disjoint Union of Paths." Axioms 14, no. 4 (2025): 293. https://doi.org/10.3390/axioms14040293.

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In this paper, we show that if a planar graph G satisfies the following conditions: (i) none of its 3-faces is adjacent to a 6−-face, and (ii) none of its 4-faces is adjacent to a 5−-face, then V(G) can be partitioned into two subsets, each induces a forest, while one of the forests has maximum degree of at most 2. This result implies that every planar graph (i) with no chordal 7−-cycles, or (ii) with neither 4- nor 6-cycles, also admits such a partition.

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VINACUA, A., I. NAVAZO, and P. BRUNET. "OCTREE DETECTION OF CLOSED COMPARTMENTS." International Journal of Computational Geometry & Applications 01, no. 03 (1991): 263–80. http://dx.doi.org/10.1142/s0218195991000190.

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The present paper addresses the problem of detecting closed compartments produced by a set of planar faces in the space. The topology of the set is general, and edges in the final piecewise planar surface can belong to one, two or more faces; boundary representations for non-manifold solids are an example. An octree structure (dubbed eightit compartment Octree) that defines a 3D graph through the volume defined by the set of faces is proposed, and it is shown that a seed propagation algorithm on the graph can be used to detect the existing closed compartments. The algorithm can either compute the total number of compartments or detect if the set of faces define a closed solid volume, the outside part being considered as a separate compartment.

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Anuradha, T., T. Lakshmi Surekha, Praveena Nuthakki, Bullarao Domathoti, Ganesh Ghorai, and Faria Ahmed Shami. "Graph Theory Algorithms of Hamiltonian Cycle from Quasi-Spanning Tree and Domination Based on Vizing Conjecture." Journal of Mathematics 2022 (August 25, 2022): 1–7. http://dx.doi.org/10.1155/2022/1618498.

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In this study, from a tree with a quasi-spanning face, the algorithm will route Hamiltonian cycles. Goodey pioneered the idea of holding facing 4 to 6 sides of a graph concurrently. Similarly, in the three connected cubic planar graphs with two-colored faces, the vertex is incident to one blue and two red faces. As a result, all red-colored faces must gain 4 to 6 sides, while all obscure-colored faces must consume 3 to 5 sides. The proposed routing approach reduces the constriction of all vertex colors and the suitable quasi-spanning tree of faces. The presented algorithm demonstrates that the spanning tree parity will determine the arbitrary face based on an even degree. As a result, when the Lemmas 1 and 2 theorems are compared, the greedy routing method of Hamiltonian cycle faces generates valuable output from a quasi-spanning tree. In graph idea, a dominating set for a graph S = V , E is a subset D of V . The range of vertices in the smallest dominating set for S is the domination number ( S ). Vizing’s conjecture from 1968 proves that the Cartesian fabricated from graphs domination variety is at least as big as their domination numbers production. Proceeding this work, the Vizing’s conjecture states that for each pair of graphs S , L .

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KRANAKIS, EVANGELOS, DANNY KRIZANC, OSCAR MORALES PONCE, and LADISLAV STACHO. "BOUNDED LENGTH, 2-EDGE AUGMENTATION OF GEOMETRIC PLANAR GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 03 (2012): 1250036. http://dx.doi.org/10.1142/s179383091250036x.

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2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any [Formula: see text].

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Sirotkin, Dmitrii V., and Dmitriy S. Malyshev. "A method of graph reduction and its applications." Discrete Mathematics and Applications 28, no. 4 (2018): 249–58. http://dx.doi.org/10.1515/dma-2018-0022.

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Abstract The independent set problem for a given simple graph is to determine the size of a maximal set of its pairwise non-adjacent vertices. We propose a new way of graph reduction leading to a new proof of the NP-completeness of the independent set problem in the class of planar graphs and to the proof of NP-completeness of this problem in the class of planar graphs having only triangular internal facets of maximal vertex degree 18.

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Alegría, Carlos, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani. "Unit-length Rectangular Drawings of Graphs." Journal of Graph Algorithms and Applications 28, no. 1 (2024): 403–37. http://dx.doi.org/10.7155/jgaa.v28i1.2996.

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A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter constraint is relaxed for the outer face. In this paper, we study rectangular drawings in which the edges have unit length. We show a complexity dichotomy for the problem of deciding the existence of a unit-length rectangular drawing, depending on whether the outer face must also be drawn as a rectangle or not. Specifically, we prove that the problem is NP-complete for biconnected graphs when the drawing of the outer face is not required to be a rectangle, even if the sought drawing must respect a given combinatorial embedding, whereas it is polynomial-time solvable, both in the fixed and the variable embedding settings, if the outer face is required to be drawn as a rectangle. Furthermore, we provide a linear-time algorithm for deciding whether a plane graph admits an embedding-preserving unit-length rectangular drawing if the drawing of the outer face is prescribed. As a by-product of our research, we provide the first polynomial-time algorithm to test whether a planar graph $G$ admits a rectangular drawing, for general instances of maximum degree $4$.

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Bienstock, Daniel, and Clyde L. Monma. "On the Complexity of Covering Vertices by Faces in a Planar Graph." SIAM Journal on Computing 17, no. 1 (1988): 53–76. http://dx.doi.org/10.1137/0217004.

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Piette, Bernard. "Bi-Symmetric Polyhedral Cages with Nearly Maximally Connected Faces and Small Holes." Symmetry 17, no. 6 (2025): 940. https://doi.org/10.3390/sym17060940.

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Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where the holes are contributed by at most four faces. We start the construction from a planar graph made out of two families of equivalent nodes. We construct the dual of the solid corresponding to that graph, and we tile its faces with regular or nearly regular polygons. We define an energy function describing the amount of irregularity of the p-cages, which we then minimise using a simulated annealing algorithm. We analyse over 600,000 possible geometries but restrict ourselves to p-cages made out of faces with deformations not exceeding 10%. We then present graphically some of the most promising geometries for protein nanocages.

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Chrobak, Marek, and Goos Kant. "Convex Grid Drawings of 3-Connected Planar Graphs." International Journal of Computational Geometry & Applications 07, no. 03 (1997): 211–23. http://dx.doi.org/10.1142/s0218195997000144.

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We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.

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Bose, Prosenjit, Stephane Durocher, Debajyoti Mondal, Maxime Peabody, Matthew Skala, and Mohammad Abdul Wahid. "Local Routing in Convex Subdivisions." International Journal of Computational Geometry & Applications 30, no. 01 (2020): 1–17. http://dx.doi.org/10.1142/s0218195920500016.

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In various wireless networking settings, node locations determine a network’s topology, allowing the network to be modelled by a geometric graph drawn in the plane. Without any additional information, local geometric routing algorithms can guarantee delivery to the target node only in restricted classes of geometric graphs, such as triangulations. In order to guarantee delivery on more general classes of geometric graphs (e.g., convex subdivisions or planar subdivisions), previous local geometric routing algorithms required [Formula: see text] state bits to be stored and passed with the message. We present the first local geometric routing algorithm using only one state bit to guarantee delivery on convex subdivisions and, when the algorithm has knowledge of the incoming port (the preceding node on the route), the first stateless local geometric routing algorithm that guarantees delivery on edge-augmented monotone subdivisions (including all convex subdivisions). We also show that [Formula: see text] state bits are necessary in planar subdivisions in which faces may have three or more reflex vertices.

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Yang, Rui, and Mingzhu Yuan. "The Structural Properties of (2, 6)-Fullerenes." Symmetry 15, no. 11 (2023): 2078. http://dx.doi.org/10.3390/sym15112078.

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A (2,6)-fullerene F is a 2-connected cubic planar graph whose faces are only 2-length and 6-length. Furthermore, it consists of exactly three 2-length faces by Euler’s formula. The (2,6)-fullerene comes from Došlić’s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Došlić showed that the (k,6)-fullerenes only exist for k=2, 3, 4, or 5, and some of the structural properties of (k,6)-fullerene for k=3, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of (2,6)-fullerene. We discover that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable (F is called n-extendable (|V(F)|≥2n+2) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (k≥1) if the deleting of any i (0≤i≤k) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2,6)-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if F−S is connected and has no perfect matchings, where F−S denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by ak(F), is the cardinality of a smallest anti−Kekulé set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti−Kekulé number 4. Further we mainly prove that there exists a (2,6)-fullerene F having fF hexagonal faces, where fF is related to the two parameters n and m.

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Ellingham, Mark N., Herbert Fleischner, Martin Kochol, and Emanuel Wenger. "Colorability of Planar Graphs with Isolated Nontriangular Faces." Graphs and Combinatorics 20, no. 4 (2004): 443–46. http://dx.doi.org/10.1007/s00373-004-0574-z.

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Albertson, Michael O., and Bojan Mohar. "Coloring Vertices and Faces of Locally Planar Graphs." Graphs and Combinatorics 22, no. 3 (2006): 289–95. http://dx.doi.org/10.1007/s00373-006-0653-4.

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Bagheri Gh., Behrooz, Tomas Feder, Herbert Fleischner, and Carlos Subi. "On Finding Hamiltonian Cycles in Barnette Graphs." Fundamenta Informaticae 188, no. 1 (2022): 1–14. http://dx.doi.org/10.3233/fi-222139.

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In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2–factor 𝒬 via (quasi) spanning trees of faces in G/𝒬 and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show that if Barnette’s Conjecture is false, then hamiltonicity in 3–connected planar cubic bipartite graphs is an NP-complete problem.

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Coolsaet, Kris, and Stan Schein. "Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra." Symmetry 10, no. 9 (2018): 382. http://dx.doi.org/10.3390/sym10090382.

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The icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and 2 E − 3 F face (torsion) equations (enforcing planarity)—and of variables ( 3 V − 6 ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron.

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Ghorai, Ganesh, and Madhumangal Pal. "Faces and dual of m-polar fuzzy planar graphs." Journal of Intelligent & Fuzzy Systems 31, no. 3 (2016): 2043–49. http://dx.doi.org/10.3233/jifs-16433.

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Cui, Ruiqi, Emil Toftegaard Gæde, Eva Rotenberg, Leif Kobbelt, and J. Andreas Bærentzen. "Surface Reconstruction Using Rotation Systems." ACM Transactions on Graphics 43, no. 6 (2024): 1–22. http://dx.doi.org/10.1145/3687956.

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Inspired by the seminal result that a graph and an associated rotation system uniquely determine the topology of a closed manifold, we propose a combinatorial method for reconstruction of surfaces from points. Our method constructs a spanning tree and a rotation system. Since the tree is trivially a planar graph, its rotation system determines a genus zero surface with a single face which we proceed to incrementally refine by inserting edges to split faces. In order to raise the genus, special handles are added in a later stage by inserting edges between different faces and thus merging them. We apply our method to a wide range of input point clouds in order to investigate its effectiveness, and we compare our method to several other surface reconstruction methods. It turns out that our approach has two specific benefits over these other methods. First, the output mesh preserves the most information from the input point cloud. Second, our method provides control over the topology of the reconstructed surface. Code is available on https://github.com/cuirq3/RsR.

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Ragoub, Lakhdar. "Molecular Descriptors Of Certain Class Of Carbon Nanocone Networks Through Quotient Graph Approach." Journal of Combinatorial Mathematics and Combinatorial Computing 120, no. 1 (2024): 301–13. http://dx.doi.org/10.61091/jcmcc120-27.

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Nanoparticles have potential applications in a wide range of fields, including electronics, medicine and material research, because of their remarkable and exceptional attributes. Carbon nanocones are planar carbon networks with mostly hexagonal faces and a few non-hexagonal faces (mostly pentagons) in the core. Two types of nanocone configurations are possible: symmetric and asymmetric, depending on where the pentagons are positioned within the structure. In addition to being a good substitute for carbon nanotubes, carbon nanocones have made an identity for themselves in a number of fields, including biosensing, electrochemical sensing, biofuel cells, supercapacitors, gas storage devices, and biomedical applications. Their astonishing chemical and physical attributes have made them well-known and widely accepted in the fields of condensed matter physics, chemistry, material science, and nanotechnology. Mathematical and chemical breakthroughs were made possible by the concept of modeling a chemical structure as a chemical graph and quantitatively analyzing the related graph using molecular descriptors. Molecular descriptors are useful in many areas of chemistry, biology, computer science, and other sciences because they allow for the analysis of chemical structures without the need for experiments. In this work, the quotient graph approach is used to establish the distance based descriptors of symmetrically configured two-pentagonal and three-pentagonal carbon nanocones.

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Ali, Shahbaz, Muhammad Khalid Mahmood, Fairouz Tchier, and F. M. O. Tawfiq. "Classification of Upper Bound Sequences of Local Fractional Metric Dimension of Rotationally Symmetric Hexagonal Planar Networks." Journal of Mathematics 2021 (February 27, 2021): 1–24. http://dx.doi.org/10.1155/2021/6613033.

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The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks. Also, we discuss the boundedness of sequences of local fractional metric dimensions. Moreover, we summarize the sequences of local fractional metric dimension by means of their graphs.

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Ali, Patrick, Peter Dankelmann, and Simon Mukwembi. "The radius of k-connected planar graphs with bounded faces." Discrete Mathematics 312, no. 24 (2012): 3636–42. http://dx.doi.org/10.1016/j.disc.2012.08.019.

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DINITZ, YEFIM, MATTHEW J. KATZ, and ROI KRAKOVSKI. "GUARDING RECTANGULAR PARTITIONS." International Journal of Computational Geometry & Applications 19, no. 06 (2009): 579–94. http://dx.doi.org/10.1142/s0218195909003131.

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A rectangular partition is a partition of a rectangle into non-overlapping rectangles, such that no four rectangles meet at a common point. A vertex guard is a guard located at a vertex of the partition (i.e., at a corner of a rectangle); it guards the rectangles that meet at this vertex. An edge guard is a guard that patrols along an edge of the partition, and is thus equivalent to two adjacent vertex guards. We consider the problem of finding a minimum-cardinality guarding set for the rectangles of the partition. For vertex guards, we prove that guarding a given subset of the rectangles is NP-hard. For edge guards, we prove that guarding all rectangles, where guards are restricted to a given subset of the edges, is NP-hard. For both results we show a reduction from vertex cover in non-bipartite 3-connected cubic planar graphs of girth greater than three. For the second NP-hardness result, we obtain a graph-theoretic result which establishes a connection between the set of faces of a plane graph of vertex degree at most three and a vertex cover for this graph. More precisely, we prove that one can assign to each internal face a distinct vertex of the cover, which lies on the face's boundary. We show that the vertices of a rectangular partition can be colored red, green, or black, such that each rectangle has all three colors on its boundary. We conjecture that the above is also true for four colors. Finally, we obtain a worst-case upper bound on the number of edge guards that are sufficient for guarding rectangular partitions with some restrictions on their structure.

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33

Bowling, Andrew, and Bryan Freyberg. "Type $(a,b,c)$ face-magic labelings of prism graphs." Utilitas Mathematica 122 (March 30, 2025): 93–108. https://doi.org/10.61091/um122-07.

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Let $G=(V,E,F)$ be a planar graph with vertex set $V$, edge set $E$, and set of faces $F.$ For nonnegative integers $a,b,$ and $c$, a type $(a,b,c)$ face-magic labeling of $G$ is an assignment of $a$ labels to each vertex, $b$ labels to each edge, and $c$ labels to each face from the set of integer labels $\{1,2,\dots a|V|+b|E|+c|F|\}$ such that each label is used exactly once, and for each $s$-sided face $f \in F,$ the sum of the label of $f$ with the labels of the vertices and edges incident with $f$ is equal to some fixed constant $\mu_s$ for every $s.$ We find necessary and sufficient conditions for every quadruple $(a,b,c,n)$ such that the $n$-prism graph $Y_n \cong K_2 \square C_n$ admits a face-magic labeling of type $(a,b,c)$.

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Abbott, H. L., D. R. Hare, and B. Zhou. "Large faces in 4-critical planar graphs with minimum degree 4." Combinatorica 15, no. 4 (1995): 455–67. http://dx.doi.org/10.1007/bf01192518.

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Owens, P. J. "Shortness parameters for planar graphs with faces of only one type." Journal of Graph Theory 9, no. 3 (1985): 381–95. http://dx.doi.org/10.1002/jgt.3190090310.

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Herráez, Borja Javier, and Eduardo Vendrell. "Segmentación de mallas 3d de edificios históricos para levantamiento arquitectónico." Virtual Archaeology Review 9, no. 18 (2018): 66. http://dx.doi.org/10.4995/var.2018.5858.

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Advances in three-dimensional (3D) acquisition systems have introduced this technology to more fields of study, such as archaeology or architecture. In the architectural field, scanning a building is one of the first possible steps from which a 3D model can be obtained and can be later used for visualisation and/or feature analysis, thanks to computer-based pattern recognition tools. The automation of these tools allows for temporal savings and has become a strong aid for professionals, so that more and more methods are developed with this objective. In this article, a method for 3D mesh segmentation focused on the representation of historic buildings is proposed. This type of buildings is characterised by having singularities and features in façades, such as doors or windows. The main objective is to recognise these features, understanding them as those parts of the model that differ from the main structure of the building. The idea is to use a recognition algorithm for planar faces that allows users to create a graph showing the connectivity between them, therefore allowing the reflection of the shape of the 3Dmodel. At a later step, this graph is matched against some pre-defined graphs that represent the patterns to look for. Each coincidence between both graphs indicate the position of one of the characteristics sought. The developed method has proved to be effective for feature detection and suitable for inclusion in architectural surveying applications.

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Guo, R. Z., T. Yu, W. X. Wang, X. M. Li, S. J. Tang, and L. F. Xie. "GENERATING LIGHTWEIGHT BUILDING MODELS WITH PRESERVED STRUCTURAL FEATURES FROM NOISY 3D MESHES." International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLVIII-1/W2-2023 (December 13, 2023): 1029–36. http://dx.doi.org/10.5194/isprs-archives-xlviii-1-w2-2023-1029-2023.

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Abstract. Obtaining building instance models directly through photogrammetry and other means results in high polygon counts and structural detail levels. Therefore, it is an important challenge to preserve the features of instant building models and generate lightweight building models from them. In this paper, we propose an improved lightweight reconstruction method for 3D building models based on planar primitives extraction, topology correction, and optimal planes selection. Improvements due to our method arise from three aspects: (1) After plane segmentation based on a simple region growing method, a second plane segmentation is performed on the building model based on the similarity of normal vectors. (2) According to the building structural characteristics, the initial plane segmentation is checked to generate new plane regions within the necessary connection regions. (3) The intersection of candidate planes is improved by enlarging the region of candidate faces, and the topological connection is improved. Furthermore, the topological relations of all planar primitives that have been optimized are recorded in an undirected graph. Finally, a watertight and manifold lightweight model is extracted from the faces of a candidate set by energy minimization. Experiments on different data sets show that the improved method is more reasonable in plane segmentation, and has superior performance in algorithm robustness and necessary structure recovery of buildings. Even when dealing with imperfect data, a watertight model is still obtained.

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Almalki, Norah, and Pawaton Kaemawichanurat. "Domination and Independent Domination in Hexagonal Systems." Mathematics 10, no. 1 (2021): 67. http://dx.doi.org/10.3390/math10010067.

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A vertex subset D of G is a dominating set if every vertex in V(G)\D is adjacent to a vertex in D. A dominating set D is independent if G[D], the subgraph of G induced by D, contains no edge. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G, and the independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. A classical work related to the relationship between γ(G) and i(G) of a graph G was established in 1978 by Allan and Laskar. They proved that every K1,3-free graph G satisfies γ(G)=i(H). Hexagonal systems (2 connected planar graphs whose interior faces are all hexagons) have been extensively studied as they are used to present bezenoid hydrocarbon structures which play an important role in organic chemistry. The domination numbers of hexagonal systems have been studied continuously since 2018 when Hutchinson et al. posted conjectures, generated from a computer program called Conjecturing, related to the domination numbers of hexagonal systems. Very recently in 2021, Bermudo et al. answered all of these conjectures. In this paper, we extend these studies by considering the relationship between the domination number and the independent domination number of hexagonal systems. Although every hexagonal system H with at least two hexagons contains K1,3 as an induced subgraph, we find many classes of hexagonal systems whose domination number is equal to an independent domination number. However, we establish the existence of a hexagonal system H such that γ(H)<i(H) with the prescribed number of hexagons.

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Xiong, B., S. Oude Elberink, and G. Vosselman. "Building modeling from noisy photogrammetric point clouds." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences II-3 (August 7, 2014): 197–204. http://dx.doi.org/10.5194/isprsannals-ii-3-197-2014.

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The Multi-View Stereo (MVS) technology has improved significantly in the last decade, providing a much denser and more accurate point cloud than before. The point cloud now becomes a valuable data for modelling the LOD2 buildings. However, it is still not accurate enough to replace the lidar point cloud. Its relative high level of noise prevents the accurate interpretation of roof faces, e.g. one planar roof face has uneven surface of points therefore is segmented into many parts. The derived roof topology graphs are quite erroneous and cannot be used to model the buildings using the current methods based on roof topology graphs. We propose a parameter-free algorithm to robustly and precisely derive roof structures and building models. The points connecting roof segments are searched and grouped as structure points and structure boundaries, accordingly presenting the roof corners and boundaries. Their geometries are computed by the plane equations of their attached roof segments. If data available, the algorithm guarantees complete building structures in noisy point clouds and meanwhile achieves global optimized models. Experiments show that, when comparing to the roof topology graph based methods, the novel algorithm achieves consistent quality for both lidar and photogrammetric point clouds. But the new method is fully automatic and is a good alternative for the model-driven method when the processing time is important.

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Wang, Junyong, Liang Chang, Hongyu Chen, and Zhencai Zhu. "Networking Feasibility of Quantum Key Distribution Constellation Networks." Entropy 24, no. 2 (2022): 298. http://dx.doi.org/10.3390/e24020298.

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Quantum key distribution constellation is the key to achieve global quantum networking. However, the networking feasibility of quantum constellation that combines satellite-to-ground accesses selection and inter-satellite routing is faced with a lack of research. In this paper, satellite-to-ground accesses selection is modeled as problems to find the longest paths in directed acyclic graphs. The inter-satellite routing is interpreted as problems to find a maximum flow in graph theory. As far as we know, the above problems are initially understood from the perspective of graph theory. Corresponding algorithms to solve the problems are provided. Although the classical discrete variable quantum key distribution protocol, i.e., BB84 protocol, is applied in simulation, the methods proposed in our paper can also be used to solve other secure key distributions. The simulation results of a low-Earth-orbit constellation scenario show that the Sun is the leading factor in restricting the networking. Due to the solar influence, inter-planar links block the network periodically and, thus, the inter-continental delivery of keys is restricted significantly.

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Chimani, Markus, Martina Juhnke-Kubitzke, and Alexander Nover. "On the bond polytope." Advances in Geometry 23, no. 4 (2023): 461–80. http://dx.doi.org/10.1515/advgeom-2023-0014.

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Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar (K 5 − e)-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on (K 5 − e)-minor free graphs.

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Collevecchio, Andrea, Abbas Mehrabian, and Nick Wormald. "Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees." Journal of Applied Probability 53, no. 3 (2016): 846–56. http://dx.doi.org/10.1017/jpr.2016.44.

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AbstractLet r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯d,t has less than tδ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t𝛿, which verifies a conjecture of Cooper and Frieze (2015).

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43

Inoue, Keisuke, Kenji Shimada, and Karthick Chilaka. "Solid Model Reconstruction of Wireframe CAD Models Based on Topological Embeddings of Planar Graphs." Journal of Mechanical Design 125, no. 3 (2003): 434–42. http://dx.doi.org/10.1115/1.1586309.

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This paper describes a robust and versatile method based on graph embedding for reconstructing a solid model from a wireframe model. The robustness and versatility of the conventional methods are limited in that: (1) most of them are heuristic and thus less robust, and (2) the rest, deterministic ones, can handle only small class of wireframes. Unlike the conventional methods, our approach is deterministic and covers a larger class of wireframes that are topologically 2-connected planar multigraphs. The class includes wireframes that can be interpreted as a closed two-manifold in multiple ways. The proposed algorithm consists of three steps: (1) all topological solutions are exhaustively generated using triconnected component decomposition; (2) the surface geometries for all the topological solutions are generated; and (3) the solutions are pruned down to geometrically valid ones. We also show the algorithm is extendable to the class of general planar multigraphs. The approach is characterized by generating the complete set of topological solutions without referring to the geometry of the wireframe, which makes the process free from geometric errors and instabilities. The algorithm is also fast, because even when there are many topological solutions, the total number of different faces is very small. The proposed approach provides a method for easy and intuitive geometric modeling as well as a conversion tool for legacy wireframes.

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44

Borodin, O. V. "Structure of neighborhoods of edges in planar graphs and simultaneous coloring of vertices, edges and faces." Mathematical Notes 53, no. 5 (1993): 483–89. http://dx.doi.org/10.1007/bf01208542.

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45

Yan, Jixing, Wanshou Jiang, and Jie Shan. "A GLOBAL SOLUTION TO TOPOLOGICAL RECONSTRUCTION OF BUILDING ROOF MODELS FROM AIRBORNE LIDAR POINT CLOUDS." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences III-3 (June 6, 2016): 379–86. http://dx.doi.org/10.5194/isprsannals-iii-3-379-2016.

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This paper presents a global solution to building roof topological reconstruction from LiDAR point clouds. Starting with segmented roof planes from building LiDAR points, a BSP (binary space partitioning) algorithm is used to partition the bounding box of the building into volumetric cells, whose geometric features and their topology are simultaneously determined. To resolve the inside/outside labelling problem of cells, a global energy function considering surface visibility and spatial regularization between adjacent cells is constructed and minimized via graph cuts. As a result, the cells are labelled as either inside or outside, where the planar surfaces between the inside and outside form the reconstructed building model. Two LiDAR data sets of Yangjiang (China) and Wuhan University (China) are used in the study. Experimental results show that the completeness of reconstructed roof planes is 87.5%. Comparing with existing data-driven approaches, the proposed approach is global. Roof faces and edges as well as their topology can be determined at one time via minimization of an energy function. Besides, this approach is robust to partial absence of roof planes and tends to reconstruct roof models with visibility-consistent surfaces.

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46

Hu, Pingbo, Yiming Miao, and Miaole Hou. "Reconstruction of Complex Roof Semantic Structures from 3D Point Clouds Using Local Convexity and Consistency." Remote Sensing 13, no. 10 (2021): 1946. http://dx.doi.org/10.3390/rs13101946.

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Three-dimensional (3D) building models are closely related to human activities in urban environments. Due to the variations in building styles and complexity in roof structures, automatically reconstructing 3D buildings with semantics and topology information still faces big challenges. In this paper, we present an automated modeling approach that can semantically decompose and reconstruct the complex building light detection and ranging (LiDAR) point clouds into simple parametric structures, and each generated structure is an unambiguous roof semantic unit without overlapping planar primitive. The proposed method starts by extracting roof planes using a multi-label energy minimization solution, followed by constructing a roof connection graph associated with proximity, similarity, and consistency attributes. Furthermore, a progressive decomposition and reconstruction algorithm is introduced to generate explicit semantic subparts and hierarchical representation of an isolated building. The proposed approach is performed on two various datasets and compared with the state-of-the-art reconstruction techniques. The experimental modeling results, including the assessment using the International Society for Photogrammetry and Remote Sensing (ISPRS) benchmark LiDAR datasets, demonstrate that the proposed modeling method can efficiently decompose complex building models into interpretable semantic structures.

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47

Yan, Jixing, Wanshou Jiang, and Jie Shan. "A GLOBAL SOLUTION TO TOPOLOGICAL RECONSTRUCTION OF BUILDING ROOF MODELS FROM AIRBORNE LIDAR POINT CLOUDS." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences III-3 (June 6, 2016): 379–86. http://dx.doi.org/10.5194/isprs-annals-iii-3-379-2016.

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This paper presents a global solution to building roof topological reconstruction from LiDAR point clouds. Starting with segmented roof planes from building LiDAR points, a BSP (binary space partitioning) algorithm is used to partition the bounding box of the building into volumetric cells, whose geometric features and their topology are simultaneously determined. To resolve the inside/outside labelling problem of cells, a global energy function considering surface visibility and spatial regularization between adjacent cells is constructed and minimized via graph cuts. As a result, the cells are labelled as either inside or outside, where the planar surfaces between the inside and outside form the reconstructed building model. Two LiDAR data sets of Yangjiang (China) and Wuhan University (China) are used in the study. Experimental results show that the completeness of reconstructed roof planes is 87.5%. Comparing with existing data-driven approaches, the proposed approach is global. Roof faces and edges as well as their topology can be determined at one time via minimization of an energy function. Besides, this approach is robust to partial absence of roof planes and tends to reconstruct roof models with visibility-consistent surfaces.

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Le Gall, Quentin, Bartłomiej Błaszczyszyn, Élie Cali, and Taoufik En-Najjary. "Continuum line-of-sight percolation on Poisson–Voronoi tessellations." Advances in Applied Probability 53, no. 2 (2021): 510–36. http://dx.doi.org/10.1017/apr.2020.69.

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AbstractIn this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

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Yen, Peter Tsung-Wen, Kelin Xia, and Siew Ann Cheong. "Understanding Changes in the Topology and Geometry of Financial Market Correlations during a Market Crash." Entropy 23, no. 9 (2021): 1211. http://dx.doi.org/10.3390/e23091211.

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In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.

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Hvoždara, Milan, and Dušan Majcin. "Geothermal heat flux anomaly due to a 3D prismoid situated in the second layer of a three-layered Earth." Contributions to Geophysics and Geodesy 43, no. 1 (2013): 39–58. http://dx.doi.org/10.2478/congeo-2013-0003.

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Abstract We present mathematical modelling of the stationary geothermal field for the three-layered earth which includes a three-dimensional perturbing body below the first layer (over the halfspace substratum). The unperturbed temperature field corresponds to the uniform vertical heat flux. The perturbing body is in the form of 3D prismoid with sloping side faces, while its upper and lower face are rectangles at the planes z = z1, z2. The theoretical formulae are based on the generalized theory of the double-layer potential and boundary integral equation (BIE). Special attention is paid to the quadrilateral prismoids bounded by planar skew faces. The numerical calculations were performed for the 3D prismoids (blocks), the thermal conductivity of which was greater than that in the ambient second layer, while the upper face of the prismoid may be in contact with the upper layer and the lower face may touch the bottom halfspace. Numerous graphs are shown for the disturbance of the temperature and heat flow distribution on the surface of the Earth or inside all three layers.

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Journal articles on the topic 'Faces In Planar Graph'

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