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Tight Bounds for Linkages in Planar Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

*In*this paper we prove this result

*for*

*planar*

*graphs*achieving g(k) = 2 O(k) . Our

*bound*is radically better than the

*bounds*known

*for*general

*graphs*. ... This fact is based on the celebrated Unique

*Linkage*Theorem, whose -very technical -proof gives a function g(k) that is responsible

*for*an immense parameter dependence

*in*the running time of the algorithm ... We thank Ken-ichi Kawarabayashi and Paul Wollan

*for*providing details on the

*bounds*

*in*[10] . ...

##
###
A lower bound for the tree-width of planar graphs with vital linkages
[article]

2010
*
arXiv
*
pre-print

We give single exponential lower

arXiv:1011.2136v1
fatcat:xodgceckpvfg3p55hli3ugjyfu
*bounds*both*for*the tree-width of*planar**graphs*with vital*linkages*, and*for*the size of the grid necessary*for*finding irrelevant vertices. ... The algorithm uses a*bound*on the tree-width of*graphs*with vital*linkages*, and deletion of irrelevant vertices. ...*In*this paper we give a lower*bound*by showing that a (2 k + 1) × (2 k + 1) grid may not suffice -even*in**planar**graphs*. Despite recent progress [8] , the quest*for*good upper*bounds*is still open. ...##
###
An upper bound on Euclidean embeddings of rigid graphs with 8 vertices
[article]

2014
*
arXiv
*
pre-print

A

arXiv:1204.6527v2
fatcat:cie6waskofajzfrca3sahzj3pe
*graph*is called (generically) rigid*in*R^d if,*for*any choice of sufficiently generic edge lengths, it can be embedded*in*R^d*in*a finite number of distinct ways, modulo rigid transformations. ... Here, we deal with the problem of determining the maximum number of*planar*Euclidean embeddings of minimally rigid*graphs*with 8 vertices, because this is the smallest unknown case*in*the plane. ... Our result is to prove an upper*bound*of 116 to the number of*planar*embeddings of minimally rigid*graphs*with 8 vertices*in*the plane. We conjecture that the*bound*of 116 is*tight*. ...##
###
Page 2587 of Mathematical Reviews Vol. , Issue 97D
[page]

1997
*
Mathematical Reviews
*

We also find

*tight*lower and upper*bounds**for*the j-*linkage*(equivalently, the j-width) of*graphs*with given numbers of vertices and edges. ... It is interesting to note that a lower*bound**for*the width of a*graph*had been found by Erdés; as we show, however, that*bound*is not*tight*. ...##
###
Irrelevant Vertices for the Planar Disjoint Paths Problem
[article]

2016
*
arXiv
*
pre-print

Our

arXiv:1310.2378v4
fatcat:px52n2lijbh6xbe762323inpdy
*bound*is radically better than the*bounds*known*for*general*graphs*. ...*In*this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of*planar**graphs*and achieves g(k)=O(k^3/2· 2^k). ... We thank Ken-ichi Kawarabayashi and Paul Wollan*for*providing details on the*bounds**in*[14] . ...##
###
Irrelevant vertices for the planar Disjoint Paths Problem

2017
*
Journal of combinatorial theory. Series B (Print)
*

Our

doi:10.1016/j.jctb.2016.10.001
fatcat:6wl64p462felvonvszpakrws44
*bound*is radically better than the*bounds*known*for*general*graphs*. ...*In*this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of*planar**graphs*and achieves g(k) = O(k 3 / 2 . 2 k ). ... This algorithm runs*in*22O(k) • n steps. Acknowledgment. We thank Ken-ichi Kawarabayashi and Paul Wollan*for*providing details on the*bounds**in*[14] . ...##
###
Algebraic methods for counting Euclidean embeddings of rigid graphs
[article]

2009
*
arXiv
*
pre-print

Moreover, our implementation yields upper

arXiv:0906.1437v2
fatcat:n5p43kegove7domhjmfmeikrgq
*bounds**for*n < 10*in*^2 and ^3, which reduce the existing gaps, and*tight**bounds*up to n=7*in*^3. ... The study of (minimally) rigid*graphs*is motivated by numerous applications, mostly*in*robotics and bioinformatics. ... V. thanks Günter Rote*for*insightful discussions on our conjecture. A. ...##
###
Guest Editors' Foreword

2017
*
Algorithmica
*

*In*"Multicuts

*in*

*Planar*and

*Bounded*-Genus

*Graphs*with

*Bounded*Number of Terminals", de Verdiére presents an FPT algorithm based on topological techniques,

*for*the minimum multicut problem on

*bounded*-genus ... by norms

*in*O(1) dimensions

*for*the complete-

*linkage*clustering algorithm, a popular method

*for*computing hierarchical clusterings. ...

##
###
Combing a Linkage in an Annulus
[article]

2022
*
arXiv
*
pre-print

A

arXiv:2207.04798v1
fatcat:arquew3onfcu5ff5ql7iicppwu
*linkage**in*a*graph*G of size k is a subgraph L of G whose connected components are k paths. The pattern of a*linkage*of size k is the set of k pairs formed by the endpoints of these paths. ... We deduce several variants of this result*in*the cases where s=0 and s>0. These variants permit the application of the unique*linkage*theorem on several path routing problems on embedded*graphs*. ...*In*fact, we prove Theorem 1*in*a more general setting. The*planarity*condition is only necessary*for*the part of the*graph**bounded*by the inner and outer cycle of C. ...##
###
The Parameterized Complexity of Graph Cyclability

2017
*
SIAM Journal on Discrete Mathematics
*

We give an FPT algorithm

doi:10.1137/141000014
fatcat:vr5valgfkbhebbtqbh4vmumyje
*for**planar**graphs*that runs*in*time 2 2 O(k 2 log k) · n 2 . Our algorithm is based on a series of*graph*theoretical results on cyclic*linkages**in**planar**graphs*. ... The cyclability of a*graph*is the maximum integer k*for*which every k vertices lie on a cycle. ... Therefore S is cyclable*in*G, as required. ⊓ ⊔ Vital cyclic*linkages**Tight*sequences. ...##
###
The Parameterized Complexity of Graph Cyclability
[chapter]

2014
*
Lecture Notes in Computer Science
*

We give an FPT algorithm

doi:10.1007/978-3-662-44777-2_41
fatcat:gvppwhyex5dx5ek37o3nsx22ea
*for**planar**graphs*that runs*in*time 2 2 O(k 2 log k) · n 2 . Our algorithm is based on a series of*graph*theoretical results on cyclic*linkages**in**planar**graphs*. ... The cyclability of a*graph*is the maximum integer k*for*which every k vertices lie on a cycle. ... Therefore S is cyclable*in*G, as required. ⊓ ⊔ Vital cyclic*linkages**Tight*sequences. ...##
###
The Parameterized Complexity of Graph Cyclability
[article]

2016
*
arXiv
*
pre-print

Our algorithm is based on a series of

arXiv:1412.3955v2
fatcat:lnn5zibtgvhl7bnu4sx4pt6u2y
*graph*-theoretical results on cyclic*linkages**in**planar**graphs*. ... On the positive side, we give an FPT algorithm*for**planar**graphs*that runs*in*time 2^2^O(k^2 k)· n^2. ... Kernelization lower*bounds**for**planar**graphs*Now we show that it is unlikely that Cyclability, parameterized by k, has a polynomial kernel when restricted to*planar**graphs*. ...##
###
Order-preserving drawings of trees with approximately optimal height (and small width)
[article]

2016
*
arXiv
*
pre-print

Finally we construct trees that require height 2pw(T)+1

arXiv:1606.02233v1
fatcat:fiatmeuutngjfgrb3k7j73gsyi
*in*all*planar*order-preserving straight-line drawings. ...*In*this paper, we study how to draw trees so that they are*planar*, straight-line and respect a given order of edges around each node. ... We also showed that '2' is*tight*if one uses the pathwidth*for*lower-*bounding*the height. ...##
###
An Exponential Time Parameterized Algorithm for Planar Disjoint Paths
[article]

2021
*
arXiv
*
pre-print

*in*single exponential time on surface-embedded

*graphs*and

*in*particular on

*planar*

*graphs*. ...

*In*this paper, we make the first step towards our quest

*for*designing a single exponential time algorithm

*for*Disjoint Paths by giving a 2^O(k^2)n^O(1)-time algorithm

*for*

*Planar*Disjoint Paths. ... [14]

*for*Disjoint Paths on directed

*planar*

*graphs*. ...

##
###
Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs
[chapter]

2012
*
Distance Geometry
*

Our implementation yields upper

doi:10.1007/978-1-4614-5128-0_2
fatcat:yprrnm2eqbbarduukkqyhj4rbq
*bounds**for*n ≤ 10*in*R 2 and R 3 , which reduce the existing gaps and lead to*tight**bounds**for*n ≤ 7*in*both R 2 and R 3 ;*in*particular, we describe the recent settlement ... Our approach also yields a new upper*bound**for*Laman*graphs*with 8 vertices, which is conjectured to be*tight*. ... and the National Science Foundation of China (under grant 61061130540)*for*the Sino-Danish Center*for*the Theory of Interactive Computation, within which part of this work was performed, and from the ...
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