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Academic literature on the topic 'Faces In Planar Graph'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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Journal articles on the topic "Faces In Planar Graph"

1

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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<em>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.</em>

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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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<p>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.</p>

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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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Dissertations / Theses on the topic "Faces In Planar Graph"

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Klemz, Boris [Verfasser]. "Facets of Planar Graph Drawing / Boris Klemz." Berlin : Freie Universität Berlin, 2020. http://d-nb.info/1221130323/34.

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Rutter, Ignaz [Verfasser], and D. [Akademischer Betreuer] Wagner. "The many faces of planarity : matching, augmentation, and embedding algorithms for planar graphs / Ignaz Rutter. Betreuer: D. Wagner." Karlsruhe : KIT-Bibliothek, 2011. http://d-nb.info/1015557848/34.

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Zhou, Hang. "Graph algorithms : network inference and planar graph optimization." Thesis, Paris, Ecole normale supérieure, 2015. http://www.theses.fr/2015ENSU0016/document.

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Cette thèse porte sur deux sujets d’algorithmique des graphes. Le premier sujet est l’inférence de réseaux. Quelle est la complexité pour déterminer un graphe inconnu à partir de requêtes de plus court chemin entre ses sommets ? Nous supposons que le graphe est de degré borné. Dans le problème de reconstruction, le but est de reconstruire le graphe ; tandis que dans le problème de vérification, le but est de vérifier qu’un graphe donné est correct. Nous développons des algorithmes probabilistes utilisant une décomposition en cellules de Voronoi. Ensuite, nous analysons des algorithmes de type glouton, et montrons qu’ils sont quasi-optimaux. Nous étudions aussi ces problèmes sur des familles particulières de graphes, démontrons des bornes inférieures, et étudions la reconstruction approximative. Le deuxième sujet est l’étude de deux problèmes d’optimisation sur les graphes planaires. Dans le problème de classification par corrélations, l’entrée est un graphe pondéré, où chaque arête a une étiquette h+i ou h-i, indiquant si ses extrémités sont ou non dans la même catégorie. Le but est de trouver une partition des sommets en catégories qui respecte au mieux les étiquettes. Dans le problème d’augmentation 2-arête-connexe, l’entrée est un graphe pondéré et un sous-ensemble R des arêtes. Le but est de trouver un sous-ensemble S des arêtes de poids minimum, tel que pour chaque arête de R, ses extrémités sont dans une composante 2-arête-connexe de l’union de R et S. Pour les graphes planaires, nous réduisons le premier problème au deuxième et montrons que les deux problèmes, bien que NP-durs, ont un schéma d’approximation en temps polynomial. Nous utilisons la technique récente de décomposition en briques<br>This thesis focuses on two topics of graph algorithms. The first topic is network inference. How efficiently can we find an unknown graph using shortest path queries between its vertices? We assume that the graph has bounded degree. In the reconstruction problem, the goal is to find the graph; and in the verification problem, the goal is to check whether a given graph is correct. We provide randomized algorithms based on a Voronoi cell decomposition. Next, we analyze greedy algorithms, and show that they are near-optimal. We also study the problems on special graph classes, prove lower bounds, and study the approximate reconstruction. The second topic is optimization in planar graphs. We study two problems. In the correlation clustering problem, the input is a weighted graph, where every edge has a label of h+i or h−i, indicating whether its endpoints are in the same category or in different categories. The goal is to find a partition of the vertices into categories that tries to respect the labels. In the two-edge-connected augmentation problem, the input is a weighted graph and a subset R of edges. The goal is to produce a minimum-weight subset S of edges, such that for every edge in R, its endpoints are two-edge-connected in the union of R and S. For planar graphs, we reduce correlation clustering to two-edge-connected augmentation, and show that both problems, although they are NP-hard, have a polynomial-time approximation scheme. We build on the brick decomposition technique developed recently

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Aurand, Eric William. "Infinite Planar Graphs." Thesis, University of North Texas, 2000. https://digital.library.unt.edu/ark:/67531/metadc2545/.

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How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.

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Kang, Mihyun. "Random planar structures and random graph processes." Doctoral thesis, [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=985516585.

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Hearon, Sean M. "PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS." CSUSB ScholarWorks, 2016. https://scholarworks.lib.csusb.edu/etd/427.

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A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.

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Ruas, Olivier. "The many faces of approximation in KNN graph computation." Thesis, Rennes 1, 2018. http://www.theses.fr/2018REN1S088/document.

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La quantité incroyable de contenu disponible dans les services en ligne rend le contenu intéressant incroyablement difficile à trouver. La manière la plus emblématique d’aider les utilisateurs consiste à faire des recommandations. Le graphe des K-plus-proches-voisins (K-Nearest-Neighbours (KNN)) connecte chaque utilisateur aux k autres utilisateurs qui lui sont les plus similaires, étant donnée une fonction de similarité. Le temps de calcul d’un graphe KNN exact est prohibitif dans les services en ligne. Les approches existantes approximent l’ensemble de candidats pour chaque voisinage pour diminuer le temps de calcul. Dans cette thèse, nous poussons plus loin la notion d’approximation : nous approximons les données de chaque utilisateur, la similarité et la localité de données. L’approche obtenue est nettement plus rapide que toutes les autres<br>The incredible quantity of available content in online services makes content of interest incredibly difficult to find. The most emblematic way to help the users is to do item recommendation. The K-Nearest-Neighbors (KNN) graph connects each user to its k most similar other users, according to a given similarity metric. The computation time of an exact KNN graph is prohibitive in online services. Existing approaches approximate the set of candidates for each user’s neighborhood to decrease the computation time. In this thesis we push farther the notion of approximation : we approximate the data of each user, the similarity and the data locality. The resulting approach clearly outperforms all the other ones

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Fowler, Thomas George. "Unique coloring of planar graphs." Diss., Georgia Institute of Technology, 1998. http://hdl.handle.net/1853/30358.

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Voigt, Konrad. "Structural Graph-based Metamodel Matching." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-81671.

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Data integration has been, and still is, a challenge for applications processing multiple heterogeneous data sources. Across the domains of schemas, ontologies, and metamodels, this imposes the need for mapping specifications, i.e. the task of discovering semantic correspondences between elements. Support for the development of such mappings has been researched, producing matching systems that automatically propose mapping suggestions. However, especially in the context of metamodel matching the result quality of state of the art matching techniques leaves room for improvement. Although the traditional approach of pair-wise element comparison works on smaller data sets, its quadratic complexity leads to poor runtime and memory performance and eventually to the inability to match, when applied on real-world data. The work presented in this thesis seeks to address these shortcomings. Thereby, we take advantage of the graph structure of metamodels. Consequently, we derive a planar graph edit distance as metamodel similarity metric and mining-based matching to make use of redundant information. We also propose a planar graph-based partitioning to cope with large-scale matching. These techniques are then evaluated using real-world mappings from SAP business integration scenarios and the MDA community. The results demonstrate improvement in quality and managed runtime and memory consumption for large-scale metamodel matching.

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Hliněnʹy, Petr. "Planar covers of graphs : Negami's conjecture." Diss., Georgia Institute of Technology, 1999. http://hdl.handle.net/1853/29449.

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Books on the topic "Faces In Planar Graph"

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1966-, Rahman Md Saidur, ed. Planar graph drawing. World Scientific, 2004.

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Nishizeki, T. Planar graphs: Theory and algorithms. North-Holland, 1988.

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Lins, Sóstenes. On the fundamental group of 3-gems and a "planar" class of 3-manifolds. Universidade Federal de Pernambuco, Centro de Ciências Exactas e da Natureza, Departamento de Matemática, 1985.

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author, Bugeaud Yann 1971, Hilberdink Titus author, and Sander Jürgen author, eds. Four faces of number theory. European Mathematical Society, 2015.

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Barg, Alexander, and O. R. Musin. Discrete geometry and algebraic combinatorics. American Mathematical Society, 2014.

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Planar graph drawing. World Scientific, 2005.

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Georgakopoulos, Agelos. Planar Cubic Cayley Graphs. American Mathematical Society, 2018.

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Nishizeki, T., and N. Chiba. Planar Graphs: Theory and Algorithms. Elsevier Science & Technology Books, 1988.

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Nishizeki, Takao, and Dr Md Saidur Rahman. Planar Graph Drawing (Lecture Notes Series on Computing). World Scientific Publishing Company, 2004.

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Choi, In-kyeong. On straight line representations of random planar graphs. 1991.

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Book chapters on the topic "Faces In Planar Graph"

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Mchedlidze, Tamara, Martin Nöllenburg, and Ignaz Rutter. "Drawing Planar Graphs with a Prescribed Inner Face." In Graph Drawing. Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-03841-4_28.

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Biedl, Therese, and Lesvia Elena Ruiz Velázquez. "Drawing Planar 3-Trees with Given Face-Areas." In Graph Drawing. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11805-0_30.

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Kleist, Linda. "Drawing Planar Graphs with Prescribed Face Areas." In Graph-Theoretic Concepts in Computer Science. Springer Berlin Heidelberg, 2016. http://dx.doi.org/10.1007/978-3-662-53536-3_14.

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Barthelemy, Marc. "Statistics of Faces and Typology of Planar Graphs." In Lecture Notes in Morphogenesis. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-20565-6_3.

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Koutsonas, Athanassios, and Dimitrios M. Thilikos. "Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms." In Graph-Theoretic Concepts in Computer Science. Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-92248-3_24.

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Kloks, Ton, C. M. Lee, and Jiping Liu. "New Algorithms for k-Face Cover, k-Feedback Vertex Set, and k-Disjoint Cycles on Plane and Planar Graphs." In Graph-Theoretic Concepts in Computer Science. Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/3-540-36379-3_25.

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Diestel, Reinhard. "Planar Graphs." In Graph Theory. Springer Berlin Heidelberg, 2017. http://dx.doi.org/10.1007/978-3-662-53622-3_4.

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Diestel, Reinhard. "Planar Graphs." In Graph Theory. Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/978-3-642-14279-6_4.

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Rahman, Md Saidur. "Planar Graphs." In Basic Graph Theory. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-49475-3_6.

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Yadav, Santosh Kumar. "Planar Graphs." In Advanced Graph Theory. Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-22562-8_3.

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Conference papers on the topic "Faces In Planar Graph"

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Yin, Yongkang, Xusheng Yang, Liming Liang, Xu Li, and Yuexian Zou. "Audio-Faces Intra-Frame Alignment with Graph Attention Networks for Active Speaker Detection." In ICASSP 2025 - 2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2025. https://doi.org/10.1109/icassp49660.2025.10889368.

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Sharifi, A., M. Vakili, M. R. Dindarloo, et al. "Graph-Based Simultaneous Localization and Calibration for Planar Cable-Driven Parallel Robots." In 2024 12th RSI International Conference on Robotics and Mechatronics (ICRoM). IEEE, 2024. https://doi.org/10.1109/icrom64545.2024.10903608.

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Warke, William D., J. Humberto Ramos, Prashant Ganesh, Kevin M. Brink, and Matthew T. Hale. "Pose Graph Optimization over Planar Unit Dual Quaternions: Improved Accuracy with Provably Convergent Riemannian Optimization." In 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2024. https://doi.org/10.1109/iros58592.2024.10802328.

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Cheng, Yuan-Po, and Ming-Jer Tsai. "Face Routing on a Non-Planar Graph." In MobiHoc'15: The Sixteenth ACM International Symposium on Mobile Ad Hoc Networking and Computing. ACM, 2015. http://dx.doi.org/10.1145/2746285.2746314.

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Loch, A., and M. Hollick. "Face the Enemy: Attack Detection for Planar Graph Routing." In 2013 Conference on Networked Systems (NetSys). IEEE, 2013. http://dx.doi.org/10.1109/netsys.2013.16.

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Zhang, Ying, Hai-Jun Su, Qizheng Liao, Shimin Wei, and Weiqing Li. "New Synthesis Approach for Expandable Polyhedral Linkages." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-35114.

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This paper presents a new synthesis approach for expandable polyhedral linkages, which are synthesized by inserting appropriate link groups into the faces of polyhedron and interconnecting them by a special composite hinges (called gusset by K. Wohlhart). The overconstrained expandable polyhedral linkages are movable with one degree of freedom (DOF).The link groups are single DOF scaling planar linkages. The gussets are multiple rotary joints whose axes converge at the corresponding vertices of the polyhedron and the number of the rotary joints equals the one of the faces which meet at the vertices. This new approach is suitable for any polyhedron whatever is regular or irregular polyhedron. To verify this new approach, the expandable regular hexahedral linkage is modeled in the SolidWorks and its mobility are studied based on screw theory and topology graph.

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Dave, Parag, and Hiroshi Sakurai. "Maximal Volume Decomposition and its Application to Feature Recognition." In ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium collocated with the ASME 1995 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/cie1995-0788.

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Abstract A method has been developed that decomposes an object having both planar and curved faces into volumes, called maximal volumes, using the halfspaces of the object. A maximal volume has as few concave edges as possible without introducing additional halfspaces. The object is first decomposed into minimal cells by extending the faces of the object. These minimal cells are then composed to form maximal volumes. The combinations of such minimal cells that result in maximal volumes are searched efficiently by examining the relationships among those minimal cells. With this decomposition method, a delta volume, which is the volume difference between the raw material and the finished part, is decomposed into maximal volumes. By subtracting maximal volumes from each other in different orders and applying graph matching to the resulting volumes, multiple interpretations of features can be generated.

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Ong, Siew-Hui, and Nornadia Zainal Abidin. "Non-hamiltonian 3-connected cubic planar graphs with only two types of faces besides 4-gons." In THE 22ND NATIONAL SYMPOSIUM ON MATHEMATICAL SCIENCES (SKSM22): Strengthening Research and Collaboration of Mathematical Sciences in Malaysia. AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4932476.

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Liu, Wei, and Kang Tai. "Computational Geometric Modeling and Unfolding of 3-D Folded Structures." In ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/detc2002/dac-34046.

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Due to environmental considerations, corrugated paperboard folded into appropriate 3-D structural shapes are increasingly being used as packaging cushions, as a substitute for those traditionally made of polymer foams. However, since paperboards are manufactured in the form of sheets, 3-D structures have to be created from these boards by folding. The design of the necessary flat layout pattern of a board that can be folded into a reasonably complex and intricate shape is a process requiring a lot of costly trial-and-error and creativity on the part of the designer. This paper describes a methodology developed to aid the designer by automatically and systematically generating many possible flat layouts that can be folded into a specified 3-D folded structure. The key to such a method is a computer representation of the topology/connectivity of the faces of the 3-D folded structure by a graph-theoretic model, and an algorithm to operate on this model to unfold and generate the geometry of the planar layout. The procedure is implemented on a computer and resulting flat layout designs have been generated for four example structures. Some of the issues concerning the types of folded structures that can and cannot be easily unfolded and the types of layouts that can and cannot be generated by the current methodology are discussed.

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Inoue, Keisuke, Kenji Shimada, and Karthick Chilaka. "Solid Model Reconstruction of Wireframe CAD Models." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/dac-21074.

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Abstract:

Abstract This paper describes an efficient method for reconstructing a solid model by generating its boundary surface from a wireframe model. This method utilizes a wireframe that is topologically a 2-connected planar multigraph, which can be interpreted as a solid in multiple ways. First the graph is decomposed using triconnected component decomposition; all solutions are then generated combinatorially. Each solution is a set of face loops representing a closed 2-manifold of genus 0. After giving a surface geometry for each face loop, the solutions are pruned according to prescribed geometric simplicity criteria. A method for extending the algorithm to deal with general disconnected wireframes is also described. This approach is characterized by not using geometry for reconstructing 2-manifold topology; geometry is used for discarding geometrically invalid solutions and prioritizing remaining ones. Therefore, negative influence of geometric errors is minimized and even wireframes having incomplete geometry can be handled. The algorithm provides an easy and intuitive method of geometric modeling as well as a conversion tool for existing wireframes.

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Reports on the topic "Faces In Planar Graph"

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Ja'Ja, Joseph, and S. R. Kosaraju. Parallel Algorithms for Planar Graph. Isomorphism and Related Problems. Defense Technical Information Center, 1986. http://dx.doi.org/10.21236/ada444434.

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