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Lower bounds for the number of edge-crossings over the spine in a topological book embedding of a graph

1999
*
Discrete Applied Mathematics
*

In

doi:10.1016/s0166-218x(99)00044-x
fatcat:4526aholg5bpnesbdzfsp6rmly
*a*topological*book*embedding*of**a**graph*,*the**graph*is drawn in*a*topological*book*by placing*the*vertices along*the*spine*of**the**book*and drawing*the*edges in*the*pages; edges are allowed to*cross**the*... This paper presents lower bounds on*the**number**of*edge-*crossings*over*the*spine for*a*variety*of**graphs*. These bounds show that*the*upper bound O(m log n) is essentially best possible. ? (M.S. ...*A**book*embedding with this restriction is*a*combinatorial*book*embedding.*The*pagenumber*of**a**graph*is*the*minimum*number**of*pages in which*the**graph*has*a*combinatorial*book*embedding. ...##
###
Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

2021
*
d CARTESIAN
*

*The*results obtained show that

*the*

*graph*Q_((1,s,b) ) (n) has 10-

*crossing*-critical and

*book*embedding

*of*

*graph*has 4-page

*book*. ...

*A*principal tool used in construction

*of*

*crossing*-critical

*graphs*are tiles. In

*the*tile concept, tiles can be arranged by gluing one tile to another in

*a*linear or circular fashion. ...

*The*

*crossing*

*number*

*of*

*a*

*graph*is

*the*minimum

*number*

*of*

*crossing*points

*of*edges in

*a*drawing

*of*in

*the*plane.

*The*

*crossing*

*number*

*of*

*a*

*graph*is often denoted as ( ). ...

##
###
Book embedding of 3-crossing-critical graphs with rational average degree between 3.5 and 4

2019
*
IOP Conference Series: Materials Science and Engineering
*

m, n) on

doi:10.1088/1757-899x/567/1/012016
fatcat:maa4wccz4bbafeunoxsnrxlfqe
*the*plane, there are exactly three pairwise edges*crossings*, and when deleting any edge*of**the**graph*,*the**number**of**crossings**of**the*remaining*graph*decreases. ... In this paper, we show that*the*minimal*number**of*pages needed to embed P(m, n) into*a**book*is three. ...*The**crossing**number*cr(G)*of**graph*G is*the*minimal*number**of**crossings*among all possible drawings*of**graph*G in*the*plane.*The**crossing**number*problem was introduced by Tutte [1] . ...##
###
Various heuristic algorithms to minimise the two-page crossing numbers of graphs

2015
*
Open Computer Science
*

We conjecture that

doi:10.1515/comp-2015-0004
fatcat:gmxkbqolvrhkrljq3xpjly435i
*the*optimal two-page drawing*of**a**graph*represents*the*planar drawing*of**the**graph*. ... It is found out that*the*new heuristics outperform*the*previously known heuristics and produce good approximations*of**the*planar*crossing**number*for severalwell-known*graph*families. ... Conclusion We designed several heuristics for*the*two-page*book**crossing*problem by combining two*of*one-page drawing algorithms (BB+ [12] and our AVSDF+ [13] ) with four twopage algorithms (SLOPE, ...##
###
Crossing-Optimal Acyclic Hamiltonian Path Completion and Its Application to Upward Topological Book Embeddings
[chapter]

2009
*
Lecture Notes in Computer Science
*

Based on this equivalence we infer for

doi:10.1007/978-3-642-00202-1_22
fatcat:6nn4jphp4reorhwnpvjwaqaazi
*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*... 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 ... Conclusions -Open Problems We have studied*the*problem*of*Acyclic-HPCCM and we have presented*a*linear time algorithm that computes*a**crossing*-optimal Acyclic HP-completion set for an OT st-digraphs G ...##
###
Page 5176 of Mathematical Reviews Vol. , Issue 96i
[page]

1996
*
Mathematical Reviews
*

An upper bound to

*the**crossing**number**of**the*complete*graph*drawn on*the*pages*of**a**book*. International Conference on*Graphs*and Hypergraphs (Varenna, 1991). J. Combin. Inform. ...*The**crossing**number*CR*(q, 7) is then defined to be*the*minimum*number**of*pairwise intersections*of*edges when K,, is drawn in*a**book*with g pages. . ...##
###
On ρ-Constrained Upward Topological Book Embeddings
[chapter]

2010
*
Lecture Notes in Computer Science
*

Then,

doi:10.1007/978-3-642-11805-0_40
fatcat:7ulewukybvfmzcn67fh5nf5xpu
*the*unique ρ-constrained upward topological*book*embedding*of*G (with*a*minimum*number**of*spine*crossings*) can be computed in O(n + α) time, where α is*the*total*number**of*spine*crossings*. ... Let G be an embedded planar st-digraph and ρ be*a*topological*numbering**of*G. ... In this poster, we present improved results with respect to*the**number**of*spine*crossings*per edge and*the*time required to compute*the**book*embedding. ...##
###
Experimental Evaluation of Book Drawing Algorithms
[article]

2017
*
arXiv
*
pre-print

*A*k-page

*book*drawing

*of*

*a*

*graph*G=(V,E) consists

*of*

*a*linear ordering

*of*its vertices along

*a*spine and an assignment

*of*each edge to one

*of*

*the*k pages, which are half-planes bounded by

*the*spine. ... In this paper, we propose

*a*comprehensive benchmark set

*of*challenging

*graph*classes for

*book*drawing algorithms and provide an extensive experimental study

*of*

*the*performance

*of*existing

*book*drawing ... Again, computing

*the*k-page

*book*

*crossing*

*number*, i.e.,

*the*minimum

*number*

*of*

*crossings*over all k-page

*book*drawings

*of*

*a*

*graph*, is an NP-hard problem [29, 34] and fixed-parameter tractable algorithms ...

##
###
Embedding a Graph into a d + 1-page Book with m logd n Edge-crossings over the Spine

2005
*
IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
*

Embedding

doi:10.1093/ietfec/e88-a.5.1136
fatcat:rla43itinnfptkgc6z7aycclnm
*a**Graph*into*a*d + 1-page*Book*with m log d n Edge-*crossings*over*the*Spine Miki MIYAUCHI †*a*) , Member SUMMARY*A*topological*book*embedding*of**a**graph*is an embedding in*a**book*that carries*the*... Enomoto showed that for any*graph*G having n vertices, there exists*a*three-page*book*embedding*of*G in which each edge*crosses**the*spine log n times. ... Theorem 3: There exists*a*d+1-page topological*book*embedding*of**a**graph*G with m log d n edge-*crossings*over*the*spine, where n is*the**number**of*vertices and m is*the**number**of*edges. ...##
###
Optimum embedding of complete graphs in books

1998
*
Discrete Mathematics
*

Embedding

doi:10.1016/s0012-365x(97)00131-3
fatcat:m7mungmtvnedrbg3w6wgngh5gi
*a**graph*in*a**book*is an arrangement*of*vertices in*a*line along*the*spine*of**the**book*and edges on*the*pages in such*a*way that edges residing on*the*same page do not*cross*. ... It is also proved that*the*set*of*complete*graphs*is*a*proper subset*of**the*set*of**graphs*for which*the*presented algorithm gives an optimal*number**of*pages. ...*The*width*of**a**book*embedding is*the*maximum*number**of*edges that*cross*any line perpendicular to*the*spine*of**the**book*on any page. ...##
###
Fewer bends point-set embedding with mapping

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

Γ is called an upward topological

doi:10.1109/icelce.2010.5700748
fatcat:qcapra5eqvc2dbxfw5q32kpuvi
*book*embedding*of*with*the*mapping Φ if*the*points in are aligned along*a*straight-line. ... Furthermore we also find an upper bound on total*number**of*bends in our drawing, which is, to*the*best*of*our knowledge,*the*first result on*the*total*number**of*bends for*the*point-set embedding problem ... Fig. 2 . 2 (*a*)*A*planar*graph*, (b)*a*topological*book*embedding*of*with*a*spine*crossing*by*the*edge (2, 5) . ...##
###
Page 3740 of Mathematical Reviews Vol. , Issue 96f
[page]

1996
*
Mathematical Reviews
*

*The*

*book*

*crossing*

*number*

*of*

*a*

*graph*G is defined as

*the*minimum

*number*

*of*edge

*crossings*when

*the*vertices

*of*G are placed on

*the*spine

*of*

*a*k-page

*book*and edges are drawn on pages, so that each edge ... Summary: “This paper introduces

*the*

*book*

*crossing*

*number*prob- lem, which can be viewed as

*a*variant

*of*

*the*well-known plane and surface

*crossing*

*number*problem or as

*a*generalization

*of*

*the*

*book*embedding ...

##
###
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth
[chapter]

2014
*
Lecture Notes in Computer Science
*

, and that computing

doi:10.1007/978-3-662-45803-7_18
fatcat:yol2zw3znzgddpi3ay5avnsmri
*the*2-page*crossing**number*is fixedparameter tractable with respect to*the*sum*of**the**number**of**crossings*and*the*treewidth*of**the*input*graph*. ... We show that computing*the*1-page*crossing**number*is fixed-parameter tractable with respect to*the**number**of**crossings*, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth ... This material is based upon work supported by*the*National Science Foundation under Grant CCF-1228639 and by*the*Office*of*Naval Research under Grant No. N00014-08-1-1015. ...##
###
Efficient deterministic algorithms for embedding graphs on books
[chapter]

1996
*
Lecture Notes in Computer Science
*

We derive deterministic polynomial time algorithms for

doi:10.1007/3-540-61332-3_149
fatcat:bu5jjouob5brxghev3px53g6tu
*book*embedding*of**a**graph*G = (V; E), jV j = n and jEj = m. ... We then use this algorithm to embed, in polynomial time, any*graph*G in O( p (G) m) pages, where (G) is*the*largest minimum degree over all subgraphs*of*G. ... Acknowledgments Research*of*Weiping Shi was supported by NSF grant MIP-9309120. ...##
###
Crossing numbers of graphs, lower bound techniques and algorithms: A survey
[chapter]

1995
*
Lecture Notes in Computer Science
*

We give

doi:10.1007/3-540-58950-3_364
fatcat:tv2kip32lfartiaifhul6vlwui
*a*survey*of*recent techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations*of**the**crossing**number*problem. ... Our aim is to emphasize*the*more general resuits or those results which have an algorithmic flavor, including*the*recent results*of**the*authors. ... For*a*drawing D*of*G on*a*k-page*book*, let vk(D) denote*the**number**of*edge*crossings*in D. Let G1 =< V1,E1 > and G~ =< V2,E2 > be two undirected*graphs*, IV1[ < [l/~[. ...
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