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[]\T1/cmr/m/n/12 This is gen-er-ally as-sumed for
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\T1/cmr/m/n/12 most no-tably knots, braids, hy-per-graphs[] [[][][]173[][][], [
][]220[][]], per-mu-ta-tions [[][][]104[][]], trop-i-cal curves [[][][]153[][][
],
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[]\T1/cmr/m/n/12 Odd cross-ing num-ber, in-de-pen-dent al-ge-braic cross-ing nu
m-ber, mono-tone cross-
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\T1/cmr/m/n/12 Both $[][]$ and $[][]$ vi-o-late what we called the em-bed-ding 
ax-iom: $[][]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 ) = [][](\OML
/cmm/m/it/12 K[]\OT1/cmr/m/n/12 ) =
 []

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[]\T1/cmr/m/n/12 Ábrego and Fernández-Merchant con-jec-ture that the \T1/cmr/m/
it/12 con-vex midrange
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[]\T1/cmr/m/n/12 Bipartite cylin-dri-cal draw-ings were in-tro-duced in Richter
 and Thomassen [[][][]589[][]]
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[]\T1/cmr/m/n/12 Melihov, Kure^^H^^Y£ik, Sel-jankin, Ti²£enko [[][][]500[][]] f
or $[][]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G; ^^Y\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/
12 and $[][]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G; ^^Y\OT1/cmr/m/n/12 )$\T1/cmr/m
/n/12 .
 []


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[]$[][]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G; ^^Y\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/1
2 can be com-puted in $\OML/cmm/m/it/12 O\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 n[]\
OT1/cmr/m/n/12 )$ \T1/cmr/m/n/12 [[][][]242[][]]. $[][]\OT1/cmr/m/n/12 (\OML/cm
m/m/it/12 G; ^^Y\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/12 is $[]$-complete [[][][]491[]
[]]
 []

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[]\T1/cmr/m/n/12 Negami [[][][]529[][][], [][]530[][]]. Also, Archdea-con, Bon-
ning-ton [[][][]48[][]], and Richter, Salazar
 []

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\T1/cmr/m/n/12 num-ber of sev-eral fam-i-lies of gen-er-al-ized Pe-tersen graph
s $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 n; k\OT1/cmr/m/n/12 ))$ \T1/cmr/m/n/12 a
re known [[][][]83[][]]. [][][][][]
 []

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[]$[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G\OT1/cmr/m/n/12 ) \U/msa/m/n/12 6 []\OT
1/cmr/m/n/12 (\OML/cmm/m/it/12 G\OT1/cmr/m/n/12 ) \U/msa/m/n/12 6 []\OT1/cmr/m/
n/12 (\OML/cmm/m/it/12 G\OT1/cmr/m/n/12 )$[][] \T1/cmr/m/n/12 (def-i-ni-tion). 
$[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G\OT1/cmr/m/n/12 ) \U/msa/m/n/12 6 \OT1/cm
r/m/n/12 4 [](\OML/cmm/m/it/12 G\OT1/cmr/m/n/12 )[]$[]
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[]\T1/cmr/m/n/12 Independent odd cross-ing num-ber, al-ge-braic cross-ing num-b
er, mono-tone cross-
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[]$[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 ) = 2[]$\T1/cmr/m/n/1
2 , and $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/
12 and $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 W[]\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/1
2 are known [[][][]279[][]];
 []

[76]
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[]$[]$\T1/cmr/m/n/12 -complete [[][][]380[][]]. It is $\OMS/cmsy/b/n/12 9[]$\T1
/cmr/m/n/12 -complete to test whether $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G\OT
1/cmr/m/n/12 ) = [](\OML/cmm/m/it/12 G\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/12 [[][][]
380[][]]. 
 []

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[]\T1/cmr/m/n/12 The study of cross-ings num-bers for VLSI lay-outs goes back t
o Leighton [[][][]477[][]], [][][][][]
 []

[92] [93] [94]
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ant, Ev-geniya Lagoda,
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 and Pe-dro Ramos.
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[96] [97] [98]
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[]\T1/cmr/m/n/12 Englebert Arnold. \T1/cmr/m/it/12 Die Anker-wick-lun-gen und A
nkerkon-struk-tio-nen der Gleichstrom-
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hrysan-thi N. Raftopoulou.
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[103]
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\T1/cmr/m/n/12 bers of Beyond-Planar Graphs Re-vis-ited''. In: \T1/cmr/m/it/12 
ArXiv e-prints \T1/cmr/m/n/12 (2021). [][]\T1/cmtt/m/n/12 arXiv:2105.12452[][]
 []

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\T1/cmr/m/n/12 duc-tion to the max-i-mum cut prob-lem''. In: \T1/cmr/m/it/12 Co
m-put-ing and com-bi-na-torics\T1/cmr/m/n/12 . Vol. 4112.
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\T1/cmr/m/n/12 Eu-ler genus and re-lated prob-lems''. In: \T1/cmr/m/it/12 ArXiv
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eck-erdt, Pavel Valtr,
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\T1/cmr/m/it/12 schools and for col-leges\T1/cmr/m/n/12 . .2. Ed-in-burgh: Adam
 and Charles Black, 1889, pp. xxiv+616
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[110]
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\T1/cmr/m/n/12 Graph Cross-ing Num-ber in Low-Degree Graphs''. In: \T1/cmr/m/it
/12 ArXiv e-prints \T1/cmr/m/n/12 abs/2202.06827
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\T1/cmr/m/n/12 known or bounded cross-ing num-bers''. In: \T1/cmr/m/it/12 ArXiv
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\T1/cmr/m/n/12 cross-ing con-stants for graphs classes''. In: \T1/cmr/m/it/12 A
rXiv e-prints \T1/cmr/m/n/12 (2018). [][]\T1/cmtt/m/n/12 arXiv:1811.08071[][]
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[112] [113] [114] [115]
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\T1/cmr/m/n/12 page book with $\OML/cmm/m/it/12 O\OT1/cmr/m/n/12 (\OML/cmm/m/it
/12 m [] n\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/12 cross-ings of edges over the spine'
'. In: \T1/cmr/m/it/12 SIAM J. Disc. Math.
 []


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\T1/cmtt/m/n/12 github . io / blog / 2013 / 04 / 08 / brief -[] history -[] of 
. html$[][] \T1/cmr/m/n/12 (last ac-cessed 9/21/2017)
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\T1/cmr/m/n/12 sec-ondary poly-topes of pla-nar point sets''. In: \T1/cmr/m/it/
12 ArXiv e-prints \T1/cmr/m/n/12 (2021). [][]\T1/cmtt/m/n/12 arXiv:2110.00544[]
[]
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rd. ``Chro-mo-Vis: Feature-
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[118]
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\T1/cmr/m/n/12 Tutte The-o-rem on the Sur-face of Genus 4''. In: \T1/cmr/m/it/1
2 Com-bi-na-tor-ica \T1/cmr/m/n/12 39.6 (2019), pp. 1267^^U
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[119] [120]
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\T1/cmr/m/n/12 Pla-nar Graphs and $\OML/cmm/m/it/12 ^^U$\T1/cmr/m/n/12 -flat Gr
aphs''. In: \T1/cmr/m/it/12 ArXiv e-prints \T1/cmr/m/n/12 (Nov. 2013). [][]\T1/
cmtt/m/n/12 arXiv:1311.0137[][]
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[121] [122] [123]
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\T1/cmr/m/it/12 fer-ence on Com-bi-na-torics, Graph The-ory, and Com-put-ing (B
a-ton Rouge, LA, 1988)\T1/cmr/m/n/12 .
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[124]
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[][]$\T1/cmtt/m/n/12 http : / / www . cs . clemson . edu / ~hedet / algorithms 
. html$[][] \T1/cmr/m/n/12 (last ac-cessed 1/9/2013)
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[125]
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[]\T1/cmr/m/n/12 Pak Tung Ho. ``The pro-jec-tive plane cross-ing num-ber of the
 cir-cu-lant graph $\OML/cmm/m/it/12 C\OT1/cmr/m/n/12 (3\OML/cmm/m/it/12 k\OT1/
cmr/m/n/12 ; \OMS/cmsy/m/n/12 f\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; k\OMS/cmsy/m
/n/12 g\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 ''.
 []

[126] [127] [128] [129] [130]
Overfull \hbox (1.95631pt too wide) in paragraph at lines 2306--2306
[]\T1/cmr/m/n/12 Marc van Krev-eld. ``Bold graph draw-ings''. In: \T1/cmr/m/it/
12 Com-pu-ta-tional Ge-om-e-try \T1/cmr/m/n/12 44.9 (2011),
 []


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[]\T1/cmr/m/n/12 Wei Lai and Pe-ter Eades. ``Re-mov-ing edge-node in-ter-sec-ti
ons in draw-ings of graphs''.
 []

[131]
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[]\T1/cmr/m/n/12 Yoonah Lee. ``On Genus $\OML/cmm/m/it/12 g$ \T1/cmr/m/n/12 Ori
-entable Cross-ing Num-bers of Small Com-plete Graphs''.
 []

[132] [133]
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\T1/cmr/m/n/12 Schaf-fer, Érico F. Xavier, Jorge Stolfi, Luer-bio Faria, and Ce
lina M. H. de Figueiredo.
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[]\T1/cmr/m/n/12 Miki Shimabara Miyauchi. ``Al-go-rithms for Em-bed-ding Graphs
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[134]
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\T1/cmr/m/it/12 fi-cient Al-go-rithms\T1/cmr/m/n/12 . Ed. by Su-sanne Al-bers, 
Hel-mut Alt, and Ste-fan Näher. Vol. 5760.
 []

[135]
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[]\T1/cmr/m/n/12 Petra Mutzel and Thomas Ziegler. ``The con-strained cross-ing 
min-i-miza-tion problem---
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\T1/cmr/m/n/12 a first ap-proach''. In: \T1/cmr/m/it/12 Op-er-a-tions Re-search
 Pro-ceed-ings 1998 (Zurich)\T1/cmr/m/n/12 . Berlin: Springer,
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[]\T1/cmr/m/n/12 Rafael Veiga Pocai. ``In-ap-prox-ima-bil-ity ra-tios for cross
-ing num-ber''. In: \T1/cmr/m/it/12 LAGOS'17---
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\T1/cmr/m/it/12 Net-work Vi-su-al-iza-tion\T1/cmr/m/n/12 . Ed. by He-len C. Pur
-chages and Ig-naz Rut-ter. Cham: Springer
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[141]
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[]\T1/cmr/m/n/12 Marcus Schae-fer, Eric Sedg-wick, and Daniel ’te-fankovi£. ``R
ec-og-niz-ing string graphs
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\T1/cmr/m/n/12 Num-bers''. In: \T1/cmr/m/it/12 ArXiv e-prints \T1/cmr/m/n/12 (N
ov. 2019). [][]\T1/cmtt/m/n/12 arXiv:1911.06403[][] \T1/cmr/m/n/12 (last ac-ces
sed 11/21/2019)
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\T1/cmr/m/n/12 Mul-ti-pli-ca-tion Graphs''. MA the-sis. Northridge: Cal-i-for-n
ia State Uni-ver-sity, Northridge,
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[147]
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\T1/cmr/m/n/12 graph $\OML/cmm/m/it/12 P\OT1/cmr/m/n/12 (3\OML/cmm/m/it/12 k; k
\OT1/cmr/m/n/12 )$ \T1/cmr/m/n/12 in the pro-jec-tive plane''. In: \T1/cmr/m/it
/12 ArXiv e-prints \T1/cmr/m/n/12 (2021). [][]\T1/cmtt/m/n/12 arXiv:2104.11640[
][]
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\T1/cmr/m/n/12 Tech-nique''. In: \T1/cmr/m/it/12 ArXiv e-prints \T1/cmr/m/n/12 
(Nov. 2014). [][]\T1/cmtt/m/n/12 arXiv:1412.0340[][] \T1/cmr/m/n/12 (last ac-ce
ssed 12/7/2020)
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\T1/cmr/m/n/12 Math-e-mat-ics Stud-ies. Am-s-ter-dam: North-Holland Pub-lish-in
g Co., 1984, pp. xiii+314
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