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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.  ...

### 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.  ...

### 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] .  ...

### 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.  ...

### Boxicity and Treewidth [article]

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.  ...

### 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.  ...

### 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).  ...

### 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.  ...

### 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.  ...

### 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.  ...

### The cubicity of hypercube graphs

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.  ...

### Geometric Representation of Graphs in Low Dimension [chapter]

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.  ...

### 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 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.  ...