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On the stab number of rectangle intersection graphs
[article]

2018
*
arXiv
*
pre-print

Tight upper bounds

arXiv:1804.06571v1
fatcat:m23w5dsxxram7ibms7bipswy2i
*on*the exact*stab**number*of split*graphs**with*boxicity*at**most*2 and block*graphs*are also given. ... We introduce the notion of*stab**number*and exact*stab**number*of*rectangle**intersection**graphs*, otherwise known as*graphs*of boxicity*at**most*2. ... In particular, we show (a) that all*rectangle**intersection**graphs*that are also split*graphs*have exact*stab**number**at**most*3 and that this bound is tight, and (b) an upper bound of log m*on*the exact ...##
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Development of Algorithm for Identification of Area for Maximum Coverage and Interference

2017
*
International Journal of Computer Applications
*

For a point set P, where no

doi:10.5120/ijca2017915323
fatcat:xl7t2a6z7nfava2hou2mwls5qu
*two*points have the same x or y coordinates, derive an upper bound*on*the size of the*stabbing*set of axis-parallel*rectangles*induced by each pair of points a,b ∈ P as the ... For a point set P in convex position, derive a lower bound*on*the size of the*stabbing*set axis-parallel*rectangles*induced by each pair of points a,b∈P as the diagonal of the*rectangles*. ... So we have seen that in the case when*two**rectangles*corresponding to adjacent edge*intersect*there does not exist third*rectangle*which can*intersect**with*both of them. ...##
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Fixed-parameter algorithms for Cochromatic Number and Disjoint Rectangle Stabbing via iterative localization

2013
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Information and Computation
*

This NP-complete problem is equivalent to deciding whether the cochromatic

doi:10.1016/j.ic.2013.08.007
fatcat:toybexwo6ranvlcvskka7aidju
*number*of a given permutation*graph**on*n vertices is*at**most*k. ... Our algorithm solves in fact a more general problem: within the mentioned running time, it decides whether the cochromatic*number*of a given perfect*graph**on*n vertices is*at**most*k. ... In particular, our algorithm partitions the n*rectangles*into*two*groups R 1 and R 2*with**at**most*n/2*rectangles*in each group and runs recursively*on*the*two*groups. ...##
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Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing
[chapter]

2010
*
Lecture Notes in Computer Science
*

In fact, we give a more general result: within the mentioned running time,

doi:10.1007/978-3-642-13731-0_32
fatcat:mstvuiwhhjgkfmnfegcx35lmmi
*one*can decide whether the cochromatic*number*of a given perfect*graph**on*n vertices is*at**most*k. ... This NP-complete problem is equivalent to deciding whether the cochromatic*number*, partitioning into the minimum*number*of cliques or independent sets, of a given permutation*graph**on*n vertices is*at*... In particular, our algorithm partitions the n*rectangles*into*two*groups R 1 and R 2*with**at**most*n/2*rectangles*in each group and runs recursively*on*the*two*groups. ...##
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Greedy is good: An experimental study on minimum clique cover and maximum independent set problems for randomly generated rectangles
[article]

2012
*
arXiv
*
pre-print

., R_n} of n randomly positioned axis parallel

arXiv:1212.0640v1
fatcat:eccdsw3ryvhn7f2ugn3obw4mcq
*rectangles*in 2D, the problem of computing the minimum clique cover (MCC) and maximum independent set (MIS) for the*intersection**graph*G( R) of the members ... Finally we will provide refined greedy algorithms based*on*a concept of simplicial*rectangle*. ... Nilson [14] proved that the*number*of geometric clique in G(R) can be*at**most*τ (2, c)φ(R) log c−1 2 (φ(R) + 1), where τ (2, c) is the Gallai*number*of the pairwise*intersecting*c-oriented polygons and ...##
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Parameterized Complexity of Stabbing Rectangles and Squares in the Plane
[chapter]

2009
*
Lecture Notes in Computer Science
*

k, select

doi:10.1007/978-3-642-00202-1_26
fatcat:7mguax2wwjgljjsoykn7ilokpe
*at**most*k of the lines such that every*rectangle*is*intersected*by*at*least*one*of the selected lines. ...*at**most**two*blocks of 1s per row. ... We thank Dániel Marx, who pointed us to the approach for proving that*Rectangle**Stabbing*is in W [1] . ...##
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Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

2013
*
Discrete & Computational Geometry
*

Several classical constructions illustrate the fact that the chromatic

doi:10.1007/s00454-013-9534-9
fatcat:sccdix27orhd5p2ucvcdfchhvq
*number*of a*graph*can be arbitrarily large compared to its clique*number*. ... However, until very recently, no such construction was known for*intersection**graphs*of geometric objects in the plane. ... Otherwise, if φ uses the same set of colors*on*these*two*families, then another color must be used*on*the diagonal D Q , and thus φ uses*at*least k colors*on*the sets*intersecting*the upper probe U P,Q ...##
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On a special class of boxicity 2 graphs
[article]

2016
*
arXiv
*
pre-print

A 2SIG is an axes-parallel

arXiv:1603.09561v1
fatcat:phzx2lo6dbardfzll7nvwny2vy
*rectangle**intersection**graph*where the*rectangles*have unit height (that is, length of the side parallel to Y-axis) and*intersects*either of the*two*fixed lines, parallel to ... We define and study a class of*graphs*, called 2-*stab*interval*graphs*(2SIG),*with*boxicity 2 which properly contains the class of interval*graphs*. ... The class of boxicity k*graphs*is the class of*graphs**with*boxicity*at**most*k. ...##
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A note on maximum independent sets in rectangle intersection graphs

2004
*
Information Processing Letters
*

Finding the maximum independent set in the

doi:10.1016/j.ipl.2003.09.019
fatcat:oeg6ftox3rb2nlmqfuk7wqmyti
*intersection**graph*of n axis-parallel*rectangles*is NP-hard. We re-examine*two*known approximation results for this problem. ... similar algorithm running in only O(n log n + n∆ k−1 ) time, where ∆ ≤ n denotes the maximum*number*of*rectangles*a point can be in. ... The*number*of iterations is*at**most*n/d, since*at*least d*rectangles*are removed per iteration. ...##
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Independent Sets of Dynamic Rectangles: Algorithms and Experiments
[article]

2020
*
arXiv
*
pre-print

We study the maximal independent set (MIS) and maximum independent set (MAX-IS) problems

arXiv:2002.07611v1
fatcat:v6zkwzunsrd7hfvqy6kozrsboa
*on*dynamic sets of O(n) axis-parallel*rectangles*, which can be modeled as dynamic*rectangle**intersection**graphs*. ... We conclude*with*an algorithm that maintains a 2-approximate MAX-IS for dynamic sets of uniform height and arbitrary width*rectangles**with*O(ωlog n) update time, where ω is the largest*number*of maximal ... :*one*query in T (C(M i−1 )), which can return*at**most*12 points and a constant*number*of queries in T (C i−1 )*with*the slab partition of P x . ...##
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Visibility Representations of Boxes in 2.5 Dimensions
[article]

2016
*
arXiv
*
pre-print

We prove that: (i) Every complete bipartite

arXiv:1608.08899v1
fatcat:nklafeuyvbcfjoxhqcsklznn74
*graph*admits a 2.5D-BR; (ii) The complete*graph*K_n admits a 2.5D-BR if and only if n ≤ 19; (iii) Every*graph**with*pathwidth*at**most*7 admits a 2.5D-BR, which ... We show that an n-vertex*graph*that admits a 2.5D-GBR has*at**most*4n - 6 √(n) edges and this bound is tight. ... RVRs can exist only for*graphs**with*thickness*at**most**two*and*with**at**most*6n − 20 edges [25] . ...##
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Covering and Packing of Rectilinear Subdivision
[article]

2018
*
arXiv
*
pre-print

(P1)

arXiv:1809.07214v1
fatcat:cpf7ifgbnbavhc7t67qjyorov4
*Stabbing*-Subdivision:*Stab*all bounded faces by selecting a minimum*number*of points in the plane. ... (P3) Dominating-Subdivision: Select a minimum size collection of faces such that any other face has a non-empty*intersection*(i.e., sharing an edge or a vertex)*with*some selected faces. ... Lemma 2 [13] Let G = (R ∪ B, E) be a bipartite planar*graph**on*red and blue vertex sets R and B, |R| ≥ 2, such that for every subset B ′ ⊆ B of size*at**most*k, where k is a large enough*number*, |N G ...##
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Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity

2018
*
International Symposium on Algorithms and Computation
*

A line segment

doi:10.4230/lipics.isaac.2018.61
dblp:conf/isaac/ChanD0SW18
fatcat:ifm52n7y25cbhnzfwdmvkpnvgm
*stabs*a*rectangle*if it*intersects*its left and its right boundary. ... Figure 1 An instance of*Stabbing*(*rectangles*)*with*an optimal solution (gray line segments). ... We want to count the*number*of*rectangles*that are*stabbed*by*at**most**two*segments in S. Consider any i and j satisfying 1 ≤ i ≤ m/2 < j ≤ m. ...##
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Witness Rectangle Graphs

2013
*
Graphs and Combinatorics
*

Finally, we conclude

doi:10.1007/s00373-013-1316-x
fatcat:qxwjhvk2krhjbk47ahgbbfmqdq
*with*some related results*on*the*number*of points required to*stab*all the*rectangles*defined by a set of n points. ... In a witness*rectangle**graph*(WRG)*on*vertex point set P*with*respect to witness point set W in the plane,*two*points x, y in P are adjacent whenever the open*rectangle**with*x and y as opposite corners ... A WRG has*at**most**two*non-trivial connected components. If there are exactly*two*, each has diameter*at**most*three. If there is*one*, its diameter is*at**most*six. ...##
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2D Fractional Cascading on Axis-aligned Planar Subdivisions
[article]

2020
*
arXiv
*
pre-print

In the problem, the input is a

arXiv:2009.05541v2
fatcat:4fzeakxnuzebdmyf3npxg3iwfa
*graph*G*with*constant degree and a set of values for every vertex of G. ... We present a*number*of upper and lower bounds which reveal that in 2D, the problem has a much richer structure. ...*on*trees of height*at**most*h 2 . ...
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