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On a special class of boxicity 2 graphs [article]

Sujoy Kumar Bhore, Dibyayan Chakraborty, Sandip Das, Sagnik Sen
2016 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1603.09561v1 fatcat:phzx2lo6dbardfzll7nvwny2vy

Poset boxicity of graphs

W.T Trotter, Douglas B West
1987 Discrete Mathematics  
In this paper, 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.  ... 
doi:10.1016/0012-365x(87)90247-0 fatcat:kh3xgphy75g2te34er2gguhhy4

Boxicity and Maximum degree [article]

L. Sunil Chandran, Mathew C. Francis, Naveen Sivadasan
2006 arXiv   pre-print
The concept 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] .  ... 
arXiv:math/0610262v1 fatcat:io3c2v3vxzh5ze3ojsqeo63f2u

Boxicity and maximum degree

L. Sunil Chandran, Mathew C. Francis, Naveen Sivadasan
2008 Journal of combinatorial theory. Series B (Print)  
The concept 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) ≤ 22 + 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.  ... 
doi:10.1016/j.jctb.2007.08.002 fatcat:qusozeplbnbappklglgclkarbi

Boxicity and Treewidth [article]

L. Sunil Chandran, Naveen Sivadasan
2005 arXiv   pre-print
In particular, we show that, if the 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.  ... 
arXiv:math/0505544v1 fatcat:2w7tfhnah5e57acba5k2gisjda

An upper bound for Cubicity in terms of Boxicity

L. Sunil Chandran, K. Ashik Mathew
2009 Discrete Mathematics  
The 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.  ... 
doi:10.1016/j.disc.2008.04.011 fatcat:tx3vdjxarfftrk2zq62ru5pplq

Characterization of the graphs with boxicity ⩽2

Martin Quest, Gerd Wegner
1990 Discrete Mathematics  
In this paper we will give 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).  ... 
doi:10.1016/0012-365x(90)90151-7 fatcat:xuhjleebljd53gcfs5i2f62r6a

An upper bound for Cubicity in terms of Boxicity [article]

L. Sunil Chandran, K. Ashik Mathew
2006 arXiv   pre-print
The 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.  ... 
arXiv:math/0605486v1 fatcat:chkcc535fjfntfb2m45dwebdam

Parameterized and Approximation Algorithms for Boxicity [article]

Abhijin Adiga and Jasine Babu and L. Sunil Chandran
2014 arXiv   pre-print
Extending the same idea in 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.  ... 
arXiv:1201.5958v3 fatcat:uxnv2cev5bcqvp27mewkqeq4fe

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

L. Sunil Chandran, Mathew C. Francis, Naveen Sivadasan
2008 Algorithmica  
In most cases, the first step usually is computing 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.  ... 
doi:10.1007/s00453-008-9163-5 fatcat:vmkcnp5mtzay3avm7cqarvrr2m

The cubicity of hypercube graphs

L. Sunil Chandran, Naveen Sivadasan
2008 Discrete Mathematics  
The parameter 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.  ... 
doi:10.1016/j.disc.2007.10.011 fatcat:6q3mx5a77nba5cq2zwfirpyhrm

Geometric Representation of Graphs in Low Dimension [chapter]

L. Sunil Chandran, Naveen Sivadasan
2006 Lecture Notes in Computer Science  
In most cases, the first step usually is computing 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.  ... 
doi:10.1007/11809678_42 fatcat:24akjbgbtrdz3bvbsrz2v3wvpm

On the Boxicity and Cubicity of Hypercubes [article]

L. Sunil Chandran, Naveen Sivadasan
2006 arXiv   pre-print
In this paper, we show that cub(H_d) = Θ(d/ d).The parameter 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.  ... 
arXiv:math/0605246v1 fatcat:goalhakkk5gcvpzstg5xlbndim

Boxicity and treewidth

L. Sunil Chandran, Naveen Sivadasan
2007 Journal of combinatorial theory. Series B (Print)  
An axis-parallel b-dimensional box is 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.  ... 
doi:10.1016/j.jctb.2006.12.004 fatcat:wj4q2bbj2jfstptaoowan6nhxa

Local boxicity [article]

Louis Esperet, Lyuben Lichev
2021 arXiv   pre-print
This extends 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.  ... 
arXiv:2012.04569v3 fatcat:5qnjnp475nb3to5xoq4jqc4blq
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