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Gap-Planar Graphs
[chapter]

2018
*
Lecture Notes in Computer Science
*

A

doi:10.1007/978-3-319-73915-1_41
fatcat:sbkzguldkvbppnqvncsjvt5bme
*graph*is k-*gap*-*planar*if it has a k-*gap*-*planar*drawing. ... Note that a*graph*is*planar*if and only if it is 0-*gap*-*planar*, and that k-*gap*-*planarity*is a monotone property: every subgraph of a k-*gap*-*planar**graph*is k-*gap*-*planar*. ... Acknowledgments This research started at the NII Shonan Meeting "Algorithmics for Beyond*Planar**Graphs*." ...##
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Minimum weight connectivity augmentation for planar straight-line graphs

2018
*
Theoretical Computer Science
*

complexity of recognizing $k$-

doi:10.1016/j.tcs.2018.05.031
fatcat:clr2edqn6rgrpjlh3thxtrf56m
*gap*-*planar**graphs*. ... We present results on the maximum density of $k$-*gap*-*planar**graphs*, their relationship to other classes of beyond-*planar**graphs*, characterization of $k$-*gap*-*planar*complete*graphs*, and the computational ... Acknowledgments This research started at the NII Shonan Meeting "Algorithmics for Beyond*Planar**Graphs*." ...##
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Diameter and spectral gap for planar graphs
[article]

2012
*
arXiv
*
pre-print

We prove that the spectral

arXiv:1204.4435v2
fatcat:ipickifdkbbldbgecczzfm5xii
*gap*of a finite*planar**graph*$X$ is bounded by $\lambda_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where $C$ depends only on the degree of $X$. ... This yields a negative answer to a question of Benjamini and Curien on the mixing times of the simple random walk on*planar**graphs*. ... In this note we investigate the relationship between the diameters diam X of finite*planar**graphs*X and their spectral*gaps*, i.e. the first non-zero eigenvalues λ 1 pXq of the associated combinatorial ...##
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Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs
[article]

2019
*
arXiv
*
pre-print

*planar*triangle-free

*graphs*and to

*planar*

*graphs*with no $4$-cycles and no $5$-cycles. ... By using known examples of non-$3$-choosable and non-$4$-choosable

*graphs*, this enables us to classify the complexity of $k$-Regular List Colouring restricted to

*planar*

*graphs*,

*planar*bipartite

*graphs*, ... Closing Complexity

*Gaps*for

*Planar*

*Graphs*Our new results, combined with known results, close a number of complexity

*gaps*for the ℓ-Regular List Colouring problem. ...

##
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Tight gaps in the cycle spectrum of 3-connected planar graphs
[article]

2020
*
arXiv
*
pre-print

For any positive integer k, define f(k) (respectively, f_3(k)) to be the minimal integer ≥ k such that every 3-connected

arXiv:2009.02503v2
fatcat:ht7w5fs3fbeq7dzde32mfo7ii4
*planar**graph*G (respectively, 3-connected cubic*planar**graph*G) of circumference ... For general 3-connected*planar**graphs*, Merker conjectured that there exists some positive integer c such that f(k) ≤ 2k + c for any positive integer k. ... Recently, it was initiated by Merker [1] to study*gaps*in the cycle spectrum of 3-connected*planar**graphs*. ...##
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Gaps in the cycle spectrum of 3-connected cubic planar graphs
[article]

2019
*
arXiv
*
pre-print

We prove that, for every natural number $k$, every sufficiently large 3-connected cubic

arXiv:1905.09101v1
fatcat:25uecptxhjakrjiggb5mhfmddu
*planar**graph*has a cycle whose length is in $[k,2k+9]$. ... We also show that this bound is close to being optimal by constructing, for every even $k\geq 4$, an infinite family of 3-connected cubic*planar**graphs*that contain no cycle whose length is in $[k,2k+1 ... The interval [3, 4] is a*gap*of every cubic*planar**graph*of girth 5. ...##
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Flow-Cut Gaps and Face Covers in Planar Graphs
[article]

2018
*
arXiv
*
pre-print

For general

arXiv:1811.02685v1
fatcat:ab47wlihn5e7bh3xa3w53vaswq
*graphs*with $k$ terminal pairs, the flow-cut*gap*is $O(\log k)$, and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. ... In particular, it has been conjectured that the flow-cut*gap*in*planar*networks is $O(1)$, while the known bounds place the*gap*somewhere between $2$ (Lee and Raghavendra, 2003) and $O(\sqrt{\log k})$ ... Introduction We present some new upper bounds on the*gap*between the concurrent flow and sparsest cut in*planar**graphs*in terms of the topology of the terminal set. ...##
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The first gap for total curvatures of planar graphs with nonnegative curvature
[article]

2017
*
arXiv
*
pre-print

Moreover, we classify the metric structures of ambient polygonal surfaces for

arXiv:1709.05309v1
fatcat:ewlcyb3iuvdjhpzoiqjskbkxpu
*planar**graphs*attaining this bound. ... We prove that the total curvature of a*planar**graph*with nonnegative combinatorial curvature is at least $\frac{1}{12}$ if it is positive. ... We call τ 1 the first*gap*of the total curvature for*planar**graphs*with nonnegative curvature. τ 1 := inf {Φ(G) : G ∈ PC For a semiplanar*graph*G = (V, E, F ) with nonnegative combinatorial curvature, ...##
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Filling the complexity gaps for colouring planar and bounded degree graphs

2019
*
Journal of Graph Theory
*

We give a complete classification of the complexity of k-Regular List Colouring restricted to

doi:10.1002/jgt.22459
fatcat:lugoxznu5ngbnebcyrxsaghpbe
*planar**graphs*,*planar*bipartite*graphs*,*planar*triangle-free*graphs*and to*planar**graphs*with no 4-cycles and ... We also give a complete classification of the complexity of this problem and a number of related colouring problems for*graphs*with bounded maximum degree. ... We use these results to fill some more complexity*gaps*by giving a complete complexity classification of a number of colouring problems for*graphs*with bounded maximum degree. ...##
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Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap
[article]

2020
*
arXiv
*
pre-print

Merker conjectured that if $k \ge 2$ is an integer and $G$ a 3-connected cubic

arXiv:2009.00423v1
fatcat:rst4bijnszelnjrgt7dk6hxxj4
*planar**graph*of circumference at least $k$, then the set of cycle lengths of $G$ must contain at least one element of the ... We obtain a*planar**graph*G that is clearly 3-connected and cubic. The circumference of A r+2 and B r is 2r + 5. ... Merker [1] recently proved that for any non-negative integer k every 3-connected cubic*planar**graph*G of circumference at least k satisfies C(G)∩[k, 2k + 9] = ∅. ...##
###
Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs
[chapter]

2016
*
Lecture Notes in Computer Science
*

We give a complete classification of the complexity of k-Regular List Colouring restricted to

doi:10.1007/978-3-319-29516-9_9
fatcat:aufsai6bmzcefe4qezbd3wbjfa
*planar**graphs*,*planar*bipartite*graphs*,*planar*triangle-free*graphs*and to*planar**graphs*with no 4-cycles and ... We also give a complete classification of the complexity of this problem and a number of related colouring problems for*graphs*with bounded maximum degree. ... We use these results to fill some more complexity*gaps*by giving a complete complexity classification of a number of colouring problems for*graphs*with bounded maximum degree. ...##
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Fan-Crossing Free Graphs and Their Relationship to other Beyond-Planar Graphs
[article]

2020
*
arXiv
*
pre-print

Both are prominent examples for beyond-

arXiv:2003.08468v2
fatcat:diek3q7zd5hmtklr22zpkr6wqe
*planar**graphs*. Further well-known beyond-*planar*classes are the $k$-*planar*, $k$-*gap*-*planar*, quasi-*planar*, and right angle crossing*graphs*. ... Thereby, we obtain*graphs*that are fan-crossing free and neither fan-crossing nor $k$-(*gap*)-*planar*. ... , and 1-*gap*-*planar**graphs*, respectively. ...##
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Gap sets for the spectra of cubic graphs

2021
*
Communications of the American Mathematical Society
*

We also show that every point in [ − 3 , 3 ) [-3,3) can be

doi:10.1090/cams/3
fatcat:7cerdedcrbg5rl6l3iko5bdso4
*gapped*by*planar*cubic*graphs*. Our results show that the study of spectra of cubic, and even*planar*cubic,*graphs*is subtle and very rich. ... We study*gaps*in the spectra of the adjacency matrices of large finite cubic*graphs*. ... Therefore any point ∈ [−3, 2√2] is*planar**gapped*by at least one of these four*graphs*. ...##
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Gap strings and spanning forests for bridge graphs of biconnected graphs

1996
*
Discrete Applied Mathematics
*

As a bonus, this algorithm yields a set of instructions to produce a

doi:10.1016/0166-218x(95)00080-b
fatcat:tyl6iutthzdihjib5ceqgmdjam
*planar*embedding of a biconnected*graph*. should one exist. ... A labeling scheme for the*gaps*of the bridges of a broken cycle C' of a biconnected*graph*G IS developed. ... the*graph*does not have a*planar*embedding. ...##
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Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs

2001
*
Electronic Journal of Combinatorics
*

is a

doi:10.37236/1588
fatcat:dcl3iqfed5atxocf6lx66b4wni
*gap*at 3. ... Let $G$ be a plane*graph*whose vertices are to be colored subject to constraints on some of the faces. ... All*graphs*and hypergraphs in this paper will be loopless, i.e. contain no edges of size 1. ...
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