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

2014
*
arXiv
*
pre-print

In this paper we affirmatively answer this question for the class

arXiv:1401.0684v1
fatcat:e335oxsxv5fdrb7adk7yo5xntu
*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

2015
*
Algorithmica
*

In this paper we affirmatively answer this question for the class

doi:10.1007/s00453-015-0016-8
fatcat:vabks2k2ebhr7g5v67uewkvt4y
*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]

2015
*
Lecture Notes in Computer Science
*

Experiments Hypothesis: There is a (maximal)

doi:10.1007/978-3-319-27261-0_11
fatcat:td5lvrcnjvbufgnnqcu72owu64
*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]

2015
*
arXiv
*
pre-print

It is known that every

arXiv:1510.05891v1
fatcat:u4g3uqajmrfufoqf6gaqwmgoku
*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]

2020
*
arXiv
*
pre-print

An

arXiv:2004.07630v1
fatcat:ucniwpne6vfapjx2vhxdfthd6y
*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

2007
*
Applied Mathematics Letters
*

bounds a face has a

doi:10.1016/j.aml.2006.08.019
fatcat:gktwisje7jdhpaodjnxc2kwyfu
*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

1991
*
Discrete Mathematics
*

We prove the following result: there exist trivalent n-vertex

doi:10.1016/0012-365x(91)90398-l
fatcat:tm6bek6ddfdt7oyjd4ir3reocq
*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]

2009
*
Lecture Notes in Computer Science
*

For the class

doi:10.1007/978-3-642-00202-1_22
fatcat:6nn4jphp4reorhwnpvjwaqaazi
*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

2010
*
International Conference on Electrical & Computer Engineering (ICECE 2010)
*

An upward point-set

doi:10.1109/icelce.2010.5700748
fatcat:qcapra5eqvc2dbxfw5q32kpuvi
*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

1979
*
Journal of combinatorial theory. Series B (Print)
*

A

doi:10.1016/0095-8956(79)90021-2
fatcat:h3uxe6q3gbb5vciiekp7tajzm4
*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]

2002
*
Lecture Notes in Computer Science
*

The equivalence between the problem

doi:10.1007/3-540-36151-0_16
fatcat:55wrpnwj55bfblk2wgmpl6gxku
*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

2013
*
Journal of Graph Algorithms and Applications
*

In this paper we study the

doi:10.7155/jgaa.00292
fatcat:5svtdpqmcjc2rjojjasowvjbze
*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

2019
*
International Symposium on Computational Geometry
*

We study k-

doi:10.4230/lipics.socg.2019.13
dblp:conf/compgeom/BinucciLGDMP19
fatcat:gitkuy2awfei7l5hlbdfwanrge
*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 [77] 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)

1987
*
Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87
*

The first algorithm in the literature for

doi:10.1145/28395.28437
dblp:conf/stoc/HeathI87
fatcat:gkamzdrdlzei5bnflrsn3nlwum
*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]

2018
*
arXiv
*
pre-print

[STACS14] described an O(n^2) time algorithm

arXiv:1801.07097v1
fatcat:uipadlhqzvd5be46qa47itydbe
*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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