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On a special class of boxicity 2 graphs
[article]

2016
*
arXiv
*
pre-print

We define and study

arXiv:1603.09561v1
fatcat:phzx2lo6dbardfzll7nvwny2vy
*a**class**of**graphs*, called*2*-stab interval*graphs*(2SIG), with*boxicity**2*which properly contains the*class**of*interval*graphs*. ... We provide*a*matrix characterization for*a*subclass*of*2SIG*graph*. ... Moreover, our*graph**class*is based*on*local structures*of**boxicity**2**graphs*in some sense. Thus, the study*of*this*class*may help us understand the structure*of**boxicity**2**graphs*in*a*better way. ...##
###
Poset boxicity of graphs

1987
*
Discrete Mathematics
*

In this paper,

doi:10.1016/0012-365x(87)90247-0
fatcat:kh3xgphy75g2te34er2gguhhy4
*a**special**class**of*posets is used to show that the poset*boxicity**of**a**graph**on*n points is at most O(log log n). ... Seheinerman defined the poset*boxicity**of**a**graph*G to be the minimum t such that G is the intersection*graph**of*intervals in some t-dimensional poset. ... First, we use*a**special**class**of*posets to show that the poset*boxicity**of**a**graph**on*n points is always at most O(log log n). consider Q. ...##
###
Boxicity and Maximum degree
[article]

2006
*
arXiv
*
pre-print

The concept

arXiv:math/0610262v1
fatcat:io3c2v3vxzh5ze3ojsqeo63f2u
*of**boxicity*finds applications in various areas such as ecology, operation research etc. We show that for any*graph*G with maximum degree Δ, (G) <*2*Δ^*2*+*2*. ... For*a**graph*G, its*boxicity*(G) is the minimum dimension d, such that G is representable as the intersection*graph**of*(axis--parallel) boxes in d--dimensional space. ... In fact, given any ∆, it is not difficult to construct*graphs**of**boxicity*Ω(∆)*on*arbitrarily large number*of*vertices, using*a*construction given by Roberts [10] . ...##
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Boxicity and maximum degree

2008
*
Journal of combinatorial theory. Series B (Print)
*

The concept

doi:10.1016/j.jctb.2007.08.002
fatcat:qusozeplbnbappklglgclkarbi
*of**boxicity*finds applications in various areas such as ecology, operation research etc. We show that for any*graph*G with maximum degree ∆, box(G) ≤*2*∆*2*+*2*. ... For*a**graph*G, its*boxicity*box(G) is the minimum dimension d, such that G is representable as the intersection*graph**of*(axis-parallel) boxes in d-dimensional space. ... There have been many attempts to estimate or bound the*boxicity**of**graph**classes*with*special*structure. In his pioneering work, F. S. ...##
###
Boxicity and Treewidth
[article]

2005
*
arXiv
*
pre-print

In particular, we show that, if the

arXiv:math/0505544v1
fatcat:2w7tfhnah5e57acba5k2gisjda
*boxicity**of**a**graph*is b >= 3, then there exists*a*simple cycle*of*length at least b-3 as well as an induced cycle*of*length at least floor*of*(log(b-*2*) to the base ... Our result leads to various interesting consequences, like bounding the*boxicity**of*many well known*graph**classes*, such as chordal*graphs*, circular arc*graphs*, AT-free*graphs*, co--comparability*graphs*... Consequences*on**Special**Classes**of**Graphs*Chordal*Graphs*Let C be*a*cycle in*a**graph*G.*A*chord*of*C is an edge*of*G joining two nodes*of*C which are not consecutive. ...##
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An upper bound for Cubicity in terms of Boxicity

2009
*
Discrete Mathematics
*

The

doi:10.1016/j.disc.2008.04.011
fatcat:tx3vdjxarfftrk2zq62ru5pplq
*boxicity**of*any*graph*G, box(G) is the minimum positive integer b such that G can be represented as the intersection*graph**of*axis-parallel*on*the real line. ... Any*graph**of*treewidth tw has cubicity at most (tw +*2*) log*2*n . ...*A*prime example*of**a**graph**class*defined in this way is the*class**of*interval*graphs*. ...##
###
Characterization of the graphs with boxicity ⩽2

1990
*
Discrete Mathematics
*

In this paper we will give

doi:10.1016/0012-365x(90)90151-7
fatcat:xuhjleebljd53gcfs5i2f62r6a
*a*combinatorial characterization*of*the*graphs*with b(G)s2, called*boxicity**2*-*graphs*, by means*of*the arrangement*of*zeros and*ones*in*special*matrices attached to the*graph*. ... Following Roberts [4] the*boxicity*b(G)*of**a**graph*G is defined as the smallest d such that G is the intersection*graph**of*boxes in Euclidean d-space, i.e. parallelepipeds with edges parallel to the coordinate ... Note, that the*class**of**boxicity**2*-*graphs*contains the*class**of*interval*graphs*. In the following the set*of*vertices*of**a**graph*G is called V(G), the set*of*edges E(G). ...##
###
An upper bound for Cubicity in terms of Boxicity
[article]

2006
*
arXiv
*
pre-print

The

arXiv:math/0605486v1
fatcat:chkcc535fjfntfb2m45dwebdam
*boxicity**of*any*graph*G, box(G) is the minimum positive integer b such that G can be represented as the intersection*graph**of*axis parallel b-dimensional boxes. ...*A*b-dimensional cube is*a*Cartesian product R_1 × R_2× ... × R_b, where each R_i (for 1 ≤ i ≤ b) is*a*closed interval*of*the form [a_i,a_i+1]*on*the real line. ...*A*prime example*of**a**graph**class*defined in this way is the*class**of*interval*graphs*. Definition 1. ...##
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Parameterized and Approximation Algorithms for Boxicity
[article]

2014
*
arXiv
*
pre-print

Extending the same idea in

arXiv:1201.5958v3
fatcat:uxnv2cev5bcqvp27mewkqeq4fe
*one**of*our algorithms, we also get an O(n√( n)/√( n)) factor approximation algorithm for computing*boxicity*and an O(n ( n)^3/*2*/√( n)) factor approximation algorithm for computing ...*Boxicity**of**a**graph*G(V, E), denoted by box(G), is the minimum integer k such that G can be represented as the intersection*graph**of*axis parallel boxes in R^k. ... FPT Algorithm to Compute*Boxicity**of*F + k 1 e − k*2*e*Graphs*In this section, we give*a*proof*of*Theorem*2*. Let G(V, E) be*a*F + k 1 e − k*2*e*graph**on*n vertices, where k 1 + k*2*= k. ...##
###
Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

2008
*
Algorithmica
*

In most cases, the first step usually is computing

doi:10.1007/s00453-008-9163-5
fatcat:vmkcnp5mtzay3avm7cqarvrr2m
*a*low dimensional box representation*of*the given*graph*. Deciding whether the*boxicity**of**a**graph*is at most*2*itself is NP-hard. ...*A*number*of*NP-hard problems are either polynomial time solvable or have much better approximation ratio*on*low*boxicity**graphs*. ... Acknowledgement We thank the anonymous referee for carefully reading the paper and pointing out*a*serious mistake in the last section*of*the first version*of*the paper. ...##
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The cubicity of hypercube graphs

2008
*
Discrete Mathematics
*

The parameter

doi:10.1016/j.disc.2007.10.011
fatcat:6q3mx5a77nba5cq2zwfirpyhrm
*boxicity*generalizes cubicity: the*boxicity*box(G)*of**a**graph*G is defined as the minimum dimension k such that G is representable as the intersection*graph**of*axis-parallel boxes in k-dimensional ... For*a**graph*G, its cubicity cub(G) is the minimum dimension k such that G is representable as the intersection*graph**of*(axis-parallel) cubes in k-dimensional space.*on*the real line.) ... Acknowledgement The first author's work was partially supported by*a*DST grant SR/S3/EECE/62/2006. ...##
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Geometric Representation of Graphs in Low Dimension
[chapter]

2006
*
Lecture Notes in Computer Science
*

In most cases, the first step usually is computing

doi:10.1007/11809678_42
fatcat:24akjbgbtrdz3bvbsrz2v3wvpm
*a*low dimensional box representation*of*the given*graph*. Deciding whether the*boxicity**of**a**graph*is at most*2*itself is NP-hard. ...*A*number*of*NP-hard problems are either polynomial time solvable or have much better approximation ratio*on*low*boxicity**graphs*. ... Researchers have also tried to bound the*boxicity**of**graph**classes*with*special*structure. Scheinerman [18] showed that the*boxicity**of*outer planar*graphs*is at most*2*. ...##
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On the Boxicity and Cubicity of Hypercubes
[article]

2006
*
arXiv
*
pre-print

In this paper, we show that cub(H_d) = Θ(d/ d).The parameter

arXiv:math/0605246v1
fatcat:goalhakkk5gcvpzstg5xlbndim
*boxicity*generalizes cubicity: the*boxicity*box(G)*of**a**graph*G is defined as the minimum dimension k such that G is representable as the intersection ... For*a**graph*G, its cubicity cub(G) is the minimum dimension k such that G is representable as the intersection*graph**of*(axis--parallel) cubes in k--dimensional space. ... There have also been attempts to estimate or bound the*boxicity**of**graph**classes*with*special*structure. Scheinerman [13] showed that the*boxicity**of*outer planar*graphs*is at most*2*. ...##
###
Boxicity and treewidth

2007
*
Journal of combinatorial theory. Series B (Print)
*

An axis-parallel b-dimensional box is

doi:10.1016/j.jctb.2006.12.004
fatcat:wj4q2bbj2jfstptaoowan6nhxa
*a*Cartesian product R 1 × R*2*× · · · × R b where R i (for 1 i b) is*a*closed interval*of*the form [*a*i , b i ]*on*the real line. ... Also, little is known about the structure imposed*on**a**graph*by its high*boxicity*. ... There have been many attempts to estimate or bound the*boxicity**of**graph**classes*with*special*structure. In his pioneering work, F.S. ...##
###
Local boxicity
[article]

2021
*
arXiv
*
pre-print

This extends

arXiv:2012.04569v3
fatcat:5qnjnp475nb3to5xoq4jqc4blq
*a*classical result*on**graphs**of**boxicity*at most*2*. ... We prove that the family*of**graphs**of*local*boxicity*at most*2*is χ-bounded, which means that the chromatic number*of*the*graphs*in this*class*can be bounded by*a*function*of*their clique number. ... The authors would like to thank Matěj Stehlík for interesting discussions and for suggesting to investigate the chromatic number*of**graphs**of*local*boxicity*at most*2*and their complements. ...
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