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Gap-Planar Graphs [chapter]

Sang Won Bae, Jean-Francois Baffier, Jinhee Chun, Peter Eades, Kord Eickmeyer, Luca Grilli, Seok-Hee Hong, Matias Korman, Fabrizio Montecchiani, Ignaz Rutter, Csaba D. Tóth
2018 Lecture Notes in Computer Science  
A 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."  ... 
doi:10.1007/978-3-319-73915-1_41 fatcat:sbkzguldkvbppnqvncsjvt5bme

Minimum weight connectivity augmentation for planar straight-line graphs

Hugo A. Akitaya, Rajasekhar Inkulu, Torrie L. Nichols, Diane L. Souvaine, Csaba D. Tóth, Charles R. Winston
2018 Theoretical Computer Science  
complexity of recognizing $k$-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."  ... 
doi:10.1016/j.tcs.2018.05.031 fatcat:clr2edqn6rgrpjlh3thxtrf56m

Diameter and spectral gap for planar graphs [article]

Larsen Louder, Juan Souto
2012 arXiv   pre-print
We prove that the spectral 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  ... 
arXiv:1204.4435v2 fatcat:ipickifdkbbldbgecczzfm5xii

Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs [article]

Konrad K. Dabrowski, Francois Dross, Matthew Johnson, Daniel Paulusma
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.  ... 
arXiv:1506.06564v5 fatcat:gpljlpxywfb3zd4hk7ebwpeihy

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

Qing Cui, On-Hei Solomon Lo
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 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.  ... 
arXiv:2009.02503v2 fatcat:ht7w5fs3fbeq7dzde32mfo7ii4

Gaps in the cycle spectrum of 3-connected cubic planar graphs [article]

Martin Merker
2019 arXiv   pre-print
We prove that, for every natural number $k$, every sufficiently large 3-connected cubic 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.  ... 
arXiv:1905.09101v1 fatcat:25uecptxhjakrjiggb5mhfmddu

Flow-Cut Gaps and Face Covers in Planar Graphs [article]

Robert Krauthgamer, James R. Lee, Havana Rika
2018 arXiv   pre-print
For general 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.  ... 
arXiv:1811.02685v1 fatcat:ab47wlihn5e7bh3xa3w53vaswq

The first gap for total curvatures of planar graphs with nonnegative curvature [article]

Bobo Hua, Yanhui Su
2017 arXiv   pre-print
Moreover, we classify the metric structures of ambient polygonal surfaces for 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,  ... 
arXiv:1709.05309v1 fatcat:ewlcyb3iuvdjhpzoiqjskbkxpu

Filling the complexity gaps for colouring planar and bounded degree graphs

Konrad K. Dabrowski, François Dross, Matthew Johnson, Daniël Paulusma
2019 Journal of Graph Theory  
We give a complete classification of the complexity of k-Regular List Colouring restricted to 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.  ... 
doi:10.1002/jgt.22459 fatcat:lugoxznu5ngbnebcyrxsaghpbe

Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap [article]

Carol T. Zamfirescu
2020 arXiv   pre-print
Merker conjectured that if $k \ge 2$ is an integer and $G$ a 3-connected cubic 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] = ∅.  ... 
arXiv:2009.00423v1 fatcat:rst4bijnszelnjrgt7dk6hxxj4

Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs [chapter]

Konrad K. Dabrowski, François Dross, Matthew Johnson, Daniël Paulusma
2016 Lecture Notes in Computer Science  
We give a complete classification of the complexity of k-Regular List Colouring restricted to 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.  ... 
doi:10.1007/978-3-319-29516-9_9 fatcat:aufsai6bmzcefe4qezbd3wbjfa

Fan-Crossing Free Graphs and Their Relationship to other Beyond-Planar Graphs [article]

Franz J. Brandenburg
2020 arXiv   pre-print
Both are prominent examples for beyond-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.  ... 
arXiv:2003.08468v2 fatcat:diek3q7zd5hmtklr22zpkr6wqe

Gap sets for the spectra of cubic graphs

Alicia Kollár, Peter Sarnak
2021 Communications of the American Mathematical Society  
We also show that every point in [ − 3 , 3 ) [-3,3) can be 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.  ... 
doi:10.1090/cams/3 fatcat:7cerdedcrbg5rl6l3iko5bdso4

Gap strings and spanning forests for bridge graphs of biconnected graphs

Peter W. Stephens
1996 Discrete Applied Mathematics  
As a bonus, this algorithm yields a set of instructions to produce a 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.  ... 
doi:10.1016/0166-218x(95)00080-b fatcat:tyl6iutthzdihjib5ceqgmdjam

Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs

Daniel Kobler, André Kündgen
2001 Electronic Journal of Combinatorics  
is a 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.  ... 
doi:10.37236/1588 fatcat:dcl3iqfed5atxocf6lx66b4wni
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