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### Two-Page Book Embeddings of 4-Planar Graphs [article]

Michael A. Bekos, Martin Gronemann, Chrysanthi N. Raftopoulou
2014 arXiv   pre-print
In this paper we affirmatively answer this question for the class of 4-planar graphs.  ...  The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book-embedding viewpoint of the problem.  ...  Two-Page Book Embeddings of General 4-Planar Graphs In this section, we prove that any planar graph of maximum degree four admits a two-page book embedding.  ...

### Two-Page Book Embeddings of 4-Planar Graphs

Michael A. Bekos, Martin Gronemann, Chrysanthi N. Raftopoulou
2015 Algorithmica
In this paper we affirmatively answer this question for the class of 4-planar graphs.  ...  The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book embedding viewpoint of the problem. ACM Subject Classification G.2.2 Graph Theory  ...  Two-Page Book Embeddings of General 4-Planar Graphs In this section, we prove that any planar graph of maximum degree 4 admits a two-page book embedding.  ...

### The Book Embedding Problem from a SAT-Solving Perspective [chapter]

Michael A. Bekos, Michael Kaufmann, Christian Zielke
2015 Lecture Notes in Computer Science
Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4.  ...  There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages.  ...  There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic.  ...

### On the Book Thickness of 1-Planar Graphs [article]

Md. Jawaherul Alam, Franz J. Brandenburg, Stephen G. Kobourov
2015 arXiv   pre-print
It is known that every planar graph has a book embedding on at most four pages. Here we investigate the book-embeddings of 1-planar graphs.  ...  We prove that every 1-planar graph has a book embedding on at most 16 pages and every 3-connected 1-planar graph has a book embedding on at most 12 pages.  ...  A 1-planar graph G has a 4-page book embedding if the planar skeleton is Hamiltonian. Proof. Let P(G) be the planar skeleton of G with Hamiltonian cycle C.  ...

### Four Pages Are Indeed Necessary for Planar Graphs [article]

Michael A. Bekos, Michael Kaufmann, Fabian Klute, Sergey Pupyrev, Chrysanthi Raftopoulou, Torsten Ueckerdt
2020 arXiv   pre-print
An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page  ...  We settle this problem by demonstrating planar graphs that require four pages in any of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.  ...  A 3-page book embedding of a graph does not contain: (i) four edges that form a 4-twist in ≺, (ii) a pair of crossing edges that both cross two edges assigned to two different pages, and (iii) an edge  ...

### Extension of a theorem of Whitney

Paul C. Kainen, Shannon Overbay
2007 Applied Mathematics Letters
bounds a face has a two-page book embedding.  ...  The latter extends a theorem of H. Whitney and gives two-page book embeddings for X-trees and square grids.  ...  For planar graphs, two pages suffice up to homeomorphism. Corollary 2. A graph is planar if and only if it is homeomorphic to a graph of book thickness at most two. Proof.  ...

### The pagewidth of trivalent planar graphs

Elena Stöhr
1991 Discrete Mathematics
We prove the following result: there exist trivalent n-vertex planar graphs, any 2-page embedding of which has pagewidth Q(n). Proof of Theorem 1. Let a 2-color circle embedding of A(k) be given.  ...  ., The pagewidth of trivalent planar graphs, Discrete Mathematics 89 (1991) 43-49.  ...  (ii) there exist 4-valent, n-vertex subhamiltonian graphs (i.e., subgraphs of planar hamiltonian graphs), any 2-page embedding of which has pagewidth Q(n).  ...

### Crossing-Optimal Acyclic Hamiltonian Path Completion and Its Application to Upward Topological Book Embeddings [chapter]

Tamara Mchedlidze, Antonios Symvonis
2009 Lecture Notes in Computer Science
For the class of planar st-digraphs, we establish an equivalence between the Acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine  ...  Based on this equivalence we infer for the class of outerplanar triangulated st-digraphs an upward topological 2-page book embedding with minimum number of spine crossings and at most one spine crossing  ...  with at most one crossing per edge of G.  ...

### Fewer bends point-set embedding with mapping

Md. Emran Chowdhury, Muhammad Jawaherul Alam, Md. Saidur Rahman
2010 International Conference on Electrical & Computer Engineering (ICECE 2010)
An upward point-set embedding of an upward planar digraph on a set of points with a mapping Φ : ( ) → is an upward planar drawing Γ of where each vertex of is placed on a point of according to Φ.  ...  We first give an algorithm that finds an upward topological book embedding of with a mapping if such an embedding exists.  ...  An upward topological book embedding of a planar graph is a drawing of on a book with two pages such that the vertices are aligned along the spine, the edges are drawn as upward on the pages and are allowed  ...

### 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  ...

### Book Embeddings and Point-Set Embeddings of Series-Parallel Digraphs [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Stephen K. Wismath
2002 Lecture Notes in Computer Science
The equivalence between the problem of computing an upward two-page book embedding and an upward point-set embedding with at most one bend per edge on any given set of points is proved.  ...  An optimal O(n)-time algorithm to compute an upward twopage book embedding of a series-parallel digraph with n vertices is presented.  ...  a two-page book embedding of a two terminal series-parallel graph.  ...

### On the Page Number of Upward Planar Directed Acyclic Graphs

Fabrizio Frati, Radoslav Fulek, Andres J. Ruiz-Vargas
2013 Journal of Graph Algorithms and Applications
In this paper we study the page number of upward planar directed acyclic graphs.  ...  We prove that: (1) the page number of any n-vertex upward planar triangulation G whose every maximal 4-connected component has page number k is at most min{O(k log n), O(2 k )}; (2) every upward planar  ...  Introduction A k-page book embedding of a graph G=(V, E) is a total ordering σ of V and a partition of E into subsets E 1 , E 2 , . . . , E k , called pages, such that no two edges (u, v) and (w, z) with  ...

### Upward Book Embeddings of st-Graphs

Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, Maurizio Patrignani, Michael Wagner
2019 International Symposium on Computational Geometry
We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of singlesource single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of  ...  Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE.  ...  Yannakakis  proved that every planar graph has a 4-page book embedding, while the fascinating question whether the page number of planar graphs can be reduced to three is still open.  ...

### The page number of genus g graphs is (g)

L. Heath, S. Istrail
1987 Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87
The first algorithm in the literature for embedding an arbitra~graph in a book with a non-trlwal upper bound on the number of pages M presented.  ...  Separate book embedding algorithms are given for the cases of graphs embedded m orlentable and nonorientable surfaces.  ...  We also thank the referees for a careful reading, for many helpful comments, for sharpening the proof of Lemma 2, and for improving the discussion of the algorithm N-LAYOUT.  ...

### Embedding 5-planar graphs in three pages [article]

Xiaxia Guan, Weihua Yang
2018 arXiv   pre-print
[STACS14] described an O(n^2) time algorithm of two-page book embedding for 4-planar graphs. In this paper, we embed 5-planar graphs into a book of three pages by an O(n^2) time algorithm.  ...  A book-embedding of a graph G is an embedding of vertices of G along the spine of a book, and edges of G on the pages so that no two edges on the same page intersect. the minimum number of pages in which  ...  [1, 2] described an O(n 2 ) algorithm to embed the 4-planar graph into a book of two pages, that is, the page number of 4-planar graphs is 2.  ...
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