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The Clique Problem in Intersection Graphs of Ellipses and Triangles

2005
*
Theory of Computing Systems
*

Here, we consider

doi:10.1007/s00224-005-1141-6
fatcat:hpicyjkvnjaldfakzdclh5r27a
*the**CLIQUE**problem*for*intersection**graphs**of**ellipses*, which,*in*a sense, interpolate between disks*and*line segments,*and*show that*the**problem*is APX -hard*in*that case. ... To our knowledge, this is*the*first hardness result for*the**CLIQUE**problem**in**intersection**graphs**of*convex objects with finite description complexity. ... Acknowledgments We thank Thomas Erlebach for introducing us to*the**CLIQUE**problem**in**intersection**graphs*,*and*for many helpful comments*and*discussions. ...##
###
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

2018
*
International Symposium on Computational Geometry
*

*In*stark contrast, Maximum

*Clique*on

*intersection*

*graphs*

*of*filled

*ellipses*or filled

*triangles*is unlikely to have such algorithms, even when

*the*

*ellipses*are close to unit disks. ... A (unit) disk

*graph*is

*the*

*intersection*

*graph*

*of*closed (unit) disks

*in*

*the*plane. ... APX-hardness was shown for Maximum

*Clique*

*in*

*the*

*intersection*

*graphs*

*of*(non-filled)

*ellipses*

*and*

*triangles*by Ambühl

*and*Wagner [4] . ...

##
###
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
[article]

2018
*
arXiv
*
pre-print

*In*stark contrast, Maximum

*Clique*on

*intersection*

*graphs*

*of*filled

*ellipses*or filled

*triangles*is unlikely to have such algorithms, even when

*the*

*ellipses*are close to unit disks. ... A (unit) disk

*graph*is

*the*

*intersection*

*graph*

*of*closed (unit) disks

*in*

*the*plane. ... APX-hardness was shown for Maximum

*Clique*

*in*

*the*

*intersection*

*graphs*

*of*(nonfilled)

*ellipses*

*and*

*triangles*by Ambühl

*and*Wagner [4] . ...

##
###
EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

2021
*
Journal of the ACM
*

*In*stark contrast, Maximum

*Clique*on ball

*graphs*

*and*unit 4-dimensional ball

*graphs*, as well as

*intersection*

*graphs*

*of*filled

*ellipses*(even close to unit disks) or filled

*triangles*is unlikely to have ... A (unit) disk

*graph*is

*the*

*intersection*

*graph*

*of*closed (unit) disks

*in*

*the*plane. ... APX-hardness was shown for Maximum

*Clique*

*in*

*the*

*intersection*

*graphs*

*of*(non-filled )

*ellipses*

*and*

*triangles*by Ambühl

*and*Wagner [7] . ...

##
###
Conservative edge sparsification for graph SLAM node removal

2014
*
2014 IEEE International Conference on Robotics and Automation (ICRA)
*

This paper reports on optimization-based methods for producing a sparse, conservative approximation

doi:10.1109/icra.2014.6906954
dblp:conf/icra/Carlevaris-BiancoE14
fatcat:364fwmqjqjhuhbsrso6bwujc7i
*of**the*dense potentials induced by node marginalization*in*simultaneous localization*and*mapping (SLAM ...*The*proposed methods start with a sparse, but overconfident, Chow-Liu tree approximation*of**the*marginalization potential*and*then use optimization-based methods to adjust*the*approximation so that it ... matrix as input as*in*[10]*and*[16] . •*The*computational complexity*of*our method is dependent only upon*the*size*of**the*elimination*clique*,*and*not on*the*size*of**the**graph*beyond*the**clique*. ...##
###
Homothetic Polygons and Beyond: Intersection Graphs, Recognition, and Maximum Clique
[article]

2016
*
arXiv
*
pre-print

We study

arXiv:1411.2928v2
fatcat:qwscl37mznc3tc2poaw7cmhs4a
*the**Clique**problem**in*classes*of**intersection**graphs**of*convex sets*in**the*plane. ...*The**problem*is known to be NP-complete*in*convex-set*intersection**graphs**and*straight-line-segment*intersection**graphs*, but solvable*in*polynomial time*in**intersection**graphs**of*homothetic*triangles*. ... Acknowledgements We thank Michael Kaufmann for valuable discussions on*the*topic. Parts*of**the*results have been presented at*the*following conferences: ISAAC 2012 [12]*and*IWCIA 2014 [6] . ...##
###
Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

2013
*
Discrete & Computational Geometry
*

However, until very recently, no such construction was known for

doi:10.1007/s00454-013-9534-9
fatcat:sccdix27orhd5p2ucvcdfchhvq
*intersection**graphs**of*geometric objects*in**the*plane. ... Several classical constructions illustrate*the*fact that*the*chromatic number*of*a*graph*can be arbitrarily large compared to its*clique*number. ... Open Access This article is distributed under*the*terms*of**the*Creative Commons Attribution License which permits any use, distribution,*and*reproduction*in*any medium, provided*the*original author(s) ...##
###
Recognizing Objects Using Color-Annotated Adjacency Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

Then

doi:10.1007/3-540-46805-6_15
fatcat:3sp2rtluond5vjai6mkbw6jtuy
*the**problem**of*recognizing*the*template object*in**the*search image is reduced to*the**problem**of*approximately matching*the*template*graph*as a subgraph*of**the*search image*graph*. ...*The*algorithm represents*the*objects*in**the*template*and*search images by weighted adjacency*graphs*. ...*the*largest*clique**in**the*association*graph*. ...##
###
Worst-case comparison of valid inequalities for the TSP

1995
*
Mathematical programming
*

*The*corresponding factor for

*the*class

*of*

*clique*tree inequalities is 8, while it is 4 for

*the*path configuration inequalities. ... We consider most

*of*

*the*known classes

*of*valid inequalities for

*the*graphical travelling salesman polyhedron

*and*compute

*the*worst-case improvement resulting from their addition to

*the*subtour polyhedron ... at least three, (5)

*the*

*intersection*

*graph*

*of*

*the*handles

*and*teeth is a tree. ...

##
###
The Clique Problem in Ray Intersection Graphs

2013
*
Discrete & Computational Geometry
*

*The*construction can be done

*in*polynomial time

*and*implies that finding a maximum

*clique*

*in*a segment

*intersection*

*graph*is NP-hard. ... Ray

*intersection*

*graphs*are

*intersection*

*graphs*

*of*rays, or halflines,

*in*

*the*plane. We show that any planar

*graph*has an even subdivision whose complement is a ray

*intersection*

*graph*. ... GReGAS

*and*ComPoSe)

*of*

*the*European Science Foundation. ...

##
###
Maximum Clique in Disk-Like Intersection Graphs
[article]

2020
*
arXiv
*
pre-print

We study

arXiv:2003.02583v1
fatcat:e5apjiz7gzgt3cj5s3ujijfjda
*the*complexity*of*Maximum*Clique**in**intersection**graphs**of*convex objects*in**the*plane. ...*The*main open question on that topic is*the*complexity*of*Maximum*Clique**in*disk*graphs*. It is not known whether this*problem*is NP-hard. ... [9] for filled*ellipses**and*filled*triangles*,*and*by Bonamy et al. [7] for ball*graphs*,*and*4-dimensional unit ball*graphs*. Bonnet et al. ...##
###
Closed trail distance in a biconnected graph

2018
*
PLoS ONE
*

*Graphs*describe

*and*represent many complex structures

*in*

*the*field

*of*social networks, biological, chemical, industrial

*and*transport systems,

*and*others. ...

*In*this article, we propose a novel metric that takes into account

*the*higher degree

*of*connectivity on

*the*part

*of*

*the*

*graph*(for example, biconnected fullerene

*graphs*

*and*fulleroids). ... Acknowledgments We would like to thank to

*the*Peter Schwerdtfeger, Lukas N. Wirz

*and*James Avery for

*the*providing

*of*

*the*fulleroid data

*and*images to be used

*in*our article. ...

##
###
Hardness Results and Efficient Algorithms for Graph Powers
[chapter]

2010
*
Lecture Notes in Computer Science
*

*The*k-th power H k

*of*a

*graph*H is obtained from H by adding new edges between every two distinct vertices having distance at most k

*in*H. ... Lau

*and*Corneil [Recognizing powers

*of*proper interval, split

*and*chordal

*graphs*, SIAM J. ...

*In*this

*and*other figures, each

*ellipse*corresponds to a

*clique*

*and*we omit

*the*

*clique*edges to keep

*the*figures simpler. ...

##
###
Grouping Curved Lines

1994
*
Procedings of the British Machine Vision Conference 1994
*

This paper examines

doi:10.5244/c.8.26
dblp:conf/bmvc/Rosin94
fatcat:2bsvryutwfbrlby6ypkalq5fa4
*the**problem**of*automatically grouping image curves.*In*contrast, most previous work has been restricted to points*and*straight lines. ... Some*of**the*computational aspects*of**the*groupings*of*continuation, parallelism,*and*proximity are analysed,*and**the*issues*of*neighbourhoods, combinatorix,*and*multiple scales are discussed. ...*The*biggest set*of*mutually parallel curves is given by*the*largest maximal*clique**in**the**graph*. ...##
###
The Complexity of Planar Counting Problems
[article]

1998
*
arXiv
*
pre-print

We prove

arXiv:cs/9809017v1
fatcat:wsh2itr3eragldlkyso7ke2ivm
*the*#P-hardness*of**the*counting*problems*associated with various satisfiability,*graph**and*combinatorial*problems*, when restricted to planar instances. ... These*problems*include 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into*Triangles*,*and**Clique*Cover. ... These suggestions significantly improved*the*quality*of*presentation*and*helped us*in*correcting a number*of*errors*in**the*earlier draft. ...
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