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Planar Graphs of Bounded Degree have Constant Queue Number
[article]

2019
*
arXiv
*
pre-print

We prove that

arXiv:1811.00816v3
fatcat:2cd65psucfe6znckdq7yps4bdm
*planar**graphs**of**bounded**degree*(which may*have*unbounded treewidth)*have**bounded**queue**number*. ... edge)*of**bounded**degree*has*bounded**queue**number*. ... Wood for pointing out an issue in an earlier version*of*this paper and the anonymous referees*of*both the journal and the conference version*of*this paper for their valuable comments and suggestions. ...##
###
Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing
[chapter]

2002
*
Lecture Notes in Computer Science
*

., 2001], who ask whether

doi:10.1007/3-540-36206-1_31
fatcat:mgvakktkh5dc7jdvby4oqnjyly
*graphs**of**bounded*tree-width*have**bounded**queue*-*number*? ... The minimum*number**of**queues*in a*queue*layout*of*a*graph*is its*queue*-*number*. Let be an Ò-vertex member*of*a proper minor-closed family*of**graphs*(such as a*planar**graph*). ... That is,*graphs**of**bounded*tree-width*have**bounded**queue*-*number*, and hence*have*three-dimensional drawings with linear volume. ...##
###
Book Embeddings of Graph Products
[article]

2020
*
arXiv
*
pre-print

We show that the stack

arXiv:2007.15102v1
fatcat:hpjrwfbyvvbtbg54hbes3wdbhe
*number*is*bounded*for the strong product*of*a path and (i) a*graph**of**bounded*pathwidth or (ii) a bipartite*graph**of**bounded*treewidth and*bounded**degree*. ... The stack*number*(book thickness, page*number*)*of*a*graph*is the minimum k such that it admits a k-stack layout. ... [2] , yield a*constant*upper*bound*on the*queue**number**of**planar**graphs*. ...##
###
Layouts of Graph Subdivisions
[chapter]

2005
*
Lecture Notes in Computer Science
*

This result reduces the question

doi:10.1007/978-3-540-31843-9_15
fatcat:kp3lauk55fhi3hibpfllpwnqta
*of*whether*queue*-*number*is*bounded*by stack-*number*to whether 3-stack*graphs**have**bounded**queue**number*. ... The*number**of*division vertices per edge in the case*of*2-*queue*and 4-track subdivisions, namely O(log qn(G)), is optimal to within a*constant*factor, for every*graph*G. ... Thanks to Franz Brandenburg and Ulrik Brandes for pointing out the connection to double-ended*queues*. ...##
###
Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seaminar 19092)

2019
*
Dagstuhl Reports
*

Next we discussed open research problems about beyond

doi:10.4230/dagrep.9.2.123
dblp:journals/dagstuhl-reports/Hong0PT19
fatcat:pvtbranvp5fwfkaul2nlmb6m4e
*planar**graphs*, such as their combinatorial structures (e.g., book thickness,*queue**number*), their topology (e.g., simultaneous embeddability, gap*planarity*... 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. ...##
###
Characterisations and examples of graph classes with bounded expansion

2012
*
European journal of combinatorics (Print)
*

*bounded*

*queue*

*number*, and

*graphs*with

*bounded*non-repetitive chromatic

*number*. ... The latter characterisation is then used to show that the notion

*of*

*bounded*expansion is compatible with Erdös-Rényi model

*of*random

*graphs*with

*constant*average

*degree*. ... Acknowledgements The authors would like to thank Vida Dujmović for simplifying a clumsy proof in an early draft

*of*the paper. ...

##
###
Queue Layouts of Graphs with Bounded Degree and Bounded Genus
[article]

2019
*
arXiv
*
pre-print

Motivated by the question

arXiv:1901.05594v2
fatcat:665twqppirgk7dhp7kubxi4gji
*of*whether*planar**graphs**have**bounded**queue*-*number*, we prove that*planar**graphs*with maximum*degree*Δ*have**queue*-*number*O(Δ^2), which improves upon the best previous*bound**of*... As a byproduct we prove that if*planar**graphs**have**bounded**queue*-*number*, then*graphs**of*Euler genus g*have**queue*-*number*O(g). ... It might even be true that*graphs*excluding a fixed minor*have**bounded**queue*-number.Planar*graphs**of**bounded**degree**have**constant**queue**number*. 2018. STOC 2019, to appear. ...##
###
Stacks, Queues and Tracks: Layouts of Graph Subdivisions

2005
*
Discrete Mathematics & Theoretical Computer Science
*

This result reduces the question

doi:10.46298/dmtcs.346
fatcat:olvfxbq3ozgapla2gms6we5srm
*of*whether*queue*-*number*is*bounded*by stack-*number*to whether 3-stack*graphs**have**bounded**queue**number*. ... The*number**of*division vertices per edge in the case*of*2-*queue*and 4-track subdivisions, namely O(log qn(G)), is optimal to within a*constant*factor, for every*graph*G. ... Thanks to Franz Brandenburg and Ulrik Brandes for pointing out the connection to double-ended*queues*. Thanks to Ferran Hurtado and Prosenjit Bose for graciously hosting the second author. ...##
###
Planar graphs have bounded queue-number
[article]

2020
*
arXiv
*
pre-print

We show that

arXiv:1904.04791v5
fatcat:pugxgwurwzcu5p3egrt45vx66u
*planar**graphs**have**bounded**queue*-*number*, thus proving a conjecture*of*Heath, Leighton and Rosenberg from 1992. ... Building on this work and using the*graph*minor structure theorem, we prove that every proper minor-closed class*of**graphs*has*bounded**queue*-*number*. ... Analogues*of*Theorems 36 and 37*have*been proved for*bounded**degree**graphs*in any minor-closed class [46] and for k-*planar**graphs*and several other non-minor-closed classes*of*interest [51] . ...##
###
Layouts of Expander Graphs
[article]

2016
*
arXiv
*
pre-print

We then show that the same

arXiv:1501.05020v3
fatcat:7v6nw5xizvgvxphqi3yipgrkeq
*graphs*admit 3-page book embeddings, 2-*queue*layouts, 4-track layouts, and*have*simple thickness 2. All these results are best possible. ... By combining this result with a generalisation*of*the unraveling method*of*Kannan, we construct 3-monotone bipartite expanders, which is best possible. ... Acknowledgement This research was initiated at the Workshop on*Graphs*and Geometry held at the Bellairs Research Institute in 2014. Thanks to the referee for the above observation. ...##
###
Graph product structure for non-minor-closed classes
[article]

2022
*
arXiv
*
pre-print

It implies, amongst other results, that k-

arXiv:1907.05168v4
fatcat:up4bm7ozlbebvetozvedvw7tiu
*planar**graphs**have*non-repetitive chromatic*number*upper-*bounded*by a function*of*k. ... ACM '20] recently proved that every*planar**graph*is isomorphic to a subgraph*of*the strong product*of*a*bounded*treewidth*graph*and a path. ... This result is the first*constant*upper*bound*on the*queue*-*number**of**bounded*powers*of**graphs*with*bounded**degree*and*bounded*Euler genus. ...##
###
Track Drawings of Graphs with Constant Queue Number
[chapter]

2004
*
Lecture Notes in Computer Science
*

In [18] it is proved that every

doi:10.1007/978-3-540-24595-7_20
fatcat:ykg7gz3imnehtkk36kyntjiicq
*graph*from a proper minor closed family has*constant*track*number*if and only if it has*constant**queue**number*. ... In this paper we study the track*number**of*well-known families*of**graphs*with small*queue**number*. ... Acknowledgments We thank Giuseppe Liotta for useful discussions on the subject*of*this paper. ...##
###
New Linear-Time Algorithms for Edge-Coloring Planar Graphs

2007
*
Algorithmica
*

We show efficient algorithms for edge-coloring

doi:10.1007/s00453-007-9044-3
fatcat:fi3i5d42jrgujfnakyd2ccweky
*planar**graphs*. Our main result is a linear-time algorithm for coloring*planar**graphs*with maximum*degree*Δ with max{Δ, 9} colors. ... Thus the coloring is optimal for*graphs*with maximum*degree*Δ ≥ 9. Moreover for Δ = 4, 5, 6 we give linear-time algorithms that use Δ + 2 colors. ... Acknowledgements The authors would like to thank anonymous referees for their excellent work and many useful suggestions which helped us avoid several mistakes and also improved the presentation*of*the ...##
###
Finding Minimum Spanning Trees

1976
*
SIAM journal on computing (Print)
*

A modification

doi:10.1137/0205051
fatcat:hgd3qksihjdflgzay7ukevgbya
*of*the method improves the l+€ running time to 0(e), if e is fl(n ) for some positive*constant*6. Another algorithm finds a minimum spanning tree*of*a*planar**graph*in 0(n) time. ... This time*bound*is the same as that*of*a new algorithm by Yao, but Yao's method seems more complicated to implement. ... For all n, there is a*graph**of*n vertices, with all vertices*of**degree*three or more,*having*a girth which is fi(log n). Finding a*planar*representation for G requires 0(e) time [5] . ...##
###
Layout of Graphs with Bounded Tree-Width

2005
*
SIAM journal on computing (Print)
*

This result provides renewed hope for the positive resolution

doi:10.1137/s0097539702416141
fatcat:lwrkyo5q7jczbel7ht6kjrgqem
*of*a*number**of*open problems in the theory*of**queue*layouts. (3) It is proved that*graphs**of**bounded*tree-width*have*three-dimensional drawings ... The minimum*number**of**queues*in a*queue*layout*of*a*graph*is its*queue*-*number*. ... Many other*graph*families*have**bounded*acyclic chromatic*number*, including*graphs*embeddable on a fixed surface [2, 3, 6] , 1-*planar**graphs*[13] ,*graphs*with*bounded*maximum*degree*[5] , and*graphs*...
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