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Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages [article]

Michael A. Bekos, Giordano Da Lozzo, Svenja Griesbach, Martin Gronemann, Fabrizio Montecchiani, Chrysanthi Raftopoulou
2022 arXiv   pre-print
For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages.  ...  An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two  ...  We also thank the useful comments of the anonymous referees of this paper; in particular, we acknowledge one referee for suggesting us to explore the relationship between k-framed and k-map graphs.  ... 
arXiv:2003.07655v2 fatcat:u547yi6d7jbhdgj7omt2ia55xu

The book thickness of a graph

Frank Bernhart, Paul C Kainen
1979 Journal of combinatorial theory. Series B (Print)  
A graph G has book thickness bt(G) < 2 if and only if it is a subgraph of a hamiltonian planar graph, but we conjecture that there are planar graphs with arbitrarily high book thickness.  ...  The book thickness bt(G) of a graph G is defined, its basic properties are delineated, and relations are given with other invariants such as thickness, genus, and chromatic number.  ...  Then a graph H is dispersable if there is a book embedding of H in a book with A(H) pages and an edge-coloring of H with Since B is dispersable, let (3 be a book embedding of B in A(B) pages and let c  ... 
doi:10.1016/0095-8956(79)90021-2 fatcat:h3uxe6q3gbb5vciiekp7tajzm4

Planarizing Graphs - A Survey and Annotated Bibliography

Annegret Liebers
2001 Journal of Graph Algorithms and Applications  
Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph.  ...  We give a survey of results about such operations and related graph parameters.  ...  The book thickness (also called pagenumber ) of a graph G is the smallest number n so that G can be embedded in a book with n pages.  ... 
doi:10.7155/jgaa.00032 fatcat:zftlx3a5jnesxff5hjymjewp5u

Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352)

Maartten Löffler, Anna Lubiw, Saul Schleimer, Erin Moriarty Wolf Chambers, Michael Wagner
2019 Dagstuhl Reports  
The seminar participants investigated problems in: knot theory, trajectory analysis, algorithmic topology, computational geometry of curves, and graph drawing, with an emphasis on how low-dimensional structures  ...  This report documents the program and the outcomes of Dagstuhl Seminar 19352 "Computation in Low-Dimensional Geometry and Topology".  ...  ., isotopic Fréchet) is really at odds with finding a "nice" morph. This is because solutions to isotopic Fréchet shrink spiraling parts of the curve infinitesimally small and then unspiral them.  ... 
doi:10.4230/dagrep.9.8.84 dblp:journals/dagstuhl-reports/LofflerLSC19 fatcat:jvf6v5zuhngi3igvpucn7miqum

Fine-Grained Location-Free Planarization in Wireless Sensor Networks

Dezun Dong, Xiangke Liao, Yunhao Liu, Xiang-Yang Li, Zhengbin Pang
2013 IEEE Transactions on Mobile Computing  
The simulation results show that our method produces high-quality planar graphs with a small stretch factor in practical large-scale networks.  ...  In particular, the majority of those location-based algorithms [7], [8] are designed under communication models of unit disk graph (UDG), and a few ones [9], [10] make an effort to construct planar graphs  ...  ACKNOWLEDGMENTS The authors are grateful for a variety of valuable comments from the anonymous reviewers. Dezun Dong  ... 
doi:10.1109/tmc.2012.57 fatcat:wmtevvq7k5fvdd4n7nnfkonmai

Fine-grained location-free planarization in wireless sensor networks

Dezun Dong, Yunhao Liu, Xiangke Liao, Xiang-Yang Li
2011 2011 Proceedings IEEE INFOCOM  
The simulation results show that our method produces high-quality planar graphs with a small stretch factor in practical large-scale networks.  ...  In particular, the majority of those location-based algorithms [7], [8] are designed under communication models of unit disk graph (UDG), and a few ones [9], [10] make an effort to construct planar graphs  ...  ACKNOWLEDGMENTS The authors are grateful for a variety of valuable comments from the anonymous reviewers. Dezun Dong  ... 
doi:10.1109/infcom.2011.5934878 dblp:conf/infocom/DongLLL11 fatcat:ghdcuffg5bcnxaeieqwtsz37xa

Graphs, Skeleta and Reconstruction of Polytopes [article]

Margaret M. Bayer
2017 arXiv   pre-print
This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Gr\"unbaum's 1967 book.  ...  A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph.  ...  Acknowledgments Thanks to Karim Adiprasito, Joseph Doolittle, Eran Nevo, Isabella Novik, Raman Sanyal, Hailun Zheng and Günter Ziegler for close reading of a draft and useful comments.  ... 
arXiv:1710.00118v2 fatcat:6dr36jkjlfhrvbvs2af4zjux5e

Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seaminar 19092)

Seok-Hee Hong, Michael Kaufmann, János Pach, Csaba D. Tóth, Michael Wagner
2019 Dagstuhl Reports  
algorithms for so-called beyond-planar graphs, i.e., non-planar graphs that admit a drawing with topological constraints such as specific types of crossings, or with some forbidden crossing patterns.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory  ...  We thank Günter Rote and Martin Gronemann for asking interesting questions that led to some of this research.  ... 
doi:10.4230/dagrep.9.2.123 dblp:journals/dagstuhl-reports/Hong0PT19 fatcat:pvtbranvp5fwfkaul2nlmb6m4e

Graph Theory with Applications

E. Kay, J. A. Bondy, U. S. R. Murty
1977 Operational Research Quarterly (1970-1977)  
This book is intended as an introduction to graph theory.  ...  Appendix I11 includes a selection of interesting graphs with special properties. These may prove to be useful in testing new conjectures.  ...  A plane graph with six faces Proof Consider an embedding G of G on the sphere; such an embedding exists by virtue of theorem 9.2.  ... 
doi:10.2307/3008805 fatcat:plgnmnhvpbdzze5sxwmgubxgae

Orthogonal Graph Drawing [chapter]

Markus Eiglsperger, Sándor P. Fekete, Gunnar W. Klau
2001 Lecture Notes in Computer Science  
Let F be the set of faces in this fixed embedding, with f 0 being the exterior face.  ...  Fig. 6 . 3 . 63 Angles around a vertex in a drawing. Fig . 6.4. A family of graphs with small angular resolution.  ...  This situation is modeled by inequalities (6.16.3) with the help of a big constant M .  ... 
doi:10.1007/3-540-44969-8_6 fatcat:nvlgt2hrm5gy5dav2bkoqm27nm

A course in the mathematics of design

Jay Kappraff
1986 Computers and Mathematics with Applications  
The sequence of topics is graph theory with application to planning a floor plan; polyhedra applied to Platonic solids; tilings of the plane with application to lattice designs; tiling of three-dimensional  ...  space and spacefilling polyhedra; similarity, proportion and the golden mean with application to architectural design; transformations; mirrors and symmetry; and vectors applied to analysis of polyhedra  ...  Professor Alan Stewart helped to shape many of the ideas of the course and has been an able collaborator in teaching it each year.  ... 
doi:10.1016/0898-1221(86)90434-7 fatcat:353kkpvcxvhvnoen4mlyeqgr5y

Spectral Partitioning, Eigenvalue Bounds, and Circle Packings for Graphs of Bounded Genus

Jonathan A. Kelner
2006 SIAM journal on computing (Print)  
Algorithms, 5 (1984), pp. 391-407] that one can find asymptotically optimal separators on these graphs if given both the graph and an embedding of it onto a low genus surface.  ...  In this paper, we address two long-standing questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [J.  ...  I would also like to thank Nathan Dunfield, Phil Bowers, and Ken Stephenson for their guidance on the theory of circle packings, and Christopher Mihelich for his help in simplifying the proof of Lemma  ... 
doi:10.1137/s0097539705447244 fatcat:tns23gkpmvgwvgrntbmx7olgri

Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus

Jonathan A. Kelner
2004 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04  
Algorithms, 5 (1984), pp. 391-407] that one can find asymptotically optimal separators on these graphs if given both the graph and an embedding of it onto a low genus surface.  ...  In this paper, we address two long-standing questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [J.  ...  I would also like to thank Nathan Dunfield, Phil Bowers, and Ken Stephenson for their guidance on the theory of circle packings, and Christopher Mihelich for his help in simplifying the proof of Lemma  ... 
doi:10.1145/1007352.1007357 dblp:conf/stoc/Kelner04 fatcat:jpelkx3j55f63ps63pwszthiei

The contact invariant in sutured Floer homology [article]

Ko Honda, William H. Kazez, Gordana Matic
2007 arXiv   pre-print
We describe an invariant of a contact 3-manifold with convex boundary as an element of Juh\'asz's sutured Floer homology.  ...  Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsv\'ath and Szab\'o. This version has some clarifications and new figures.  ...  We also thank John Etnyre for the discussions which led to the reformulation of the contact invariant in [HKM2] .  ... 
arXiv:0705.2828v2 fatcat:s5nxwqssyzeadbqeui3san4giu

A SURVEY OF PROGRESS IN GRAPH THEORY IN THE SOVIET UNION

James Turner, William H. Kautz
1970 Annals of the New York Academy of Sciences  
The best Sovic work has been concerned with bounds on numerical indices associated with graphs, properties of algebraic stvuctures associated with graphs, and operations on graphs.  ...  use of graphs for modeling in social sciences.  ...  Papers of the second type deal with questions of either embedding a given graph in a graph of some special type or embedding metrics in graphs of a certain tyl)L-.  ... 
doi:10.1111/j.1749-6632.1970.tb56496.x fatcat:cbb5x22w2farrnw5sijyovn66i
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