Boxicity and maximum degree

Chandran, L. Sunil; Francis, Mathew C.; Sivadasan, Naveen

doi:10.1016/j.jctb.2007.08.002

Cited by 47 publications

(44 citation statements)

References 8 publications

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“…Roberts, in his seminal work [24] proved that the boxicity of a complete k-partite graph is k. Chandran and Sivadasan [6] showed that box(G) ≤ tree-width (G) + 2. Chandran, Francis and Sivadasan [5] proved that box(G) ≤ χ(G 2 ) where, χ(G 2 ) is the chromatic number of G 2 . In [14] Esperet proved that box(G) ≤ ∆ 2 (G) + 2, where ∆(G) is the maximum degree of G. Scheinerman [25] showed that the boxicity of outer planar graphs is at most 2.…”

Section: Boxicitymentioning

confidence: 99%

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Boxicity and Poset Dimension

Adiga

Bhowmick

Chandran

2010

Lecture Notes in Computer Science

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Abstract. Let G be a simple, undirected, finite graph with vertex set V (G) and edge set E(G). A kdimensional box is a Cartesian product of closed intervals [a1, b1] × [a2, b2] × · · · × [a k , b k ]. The boxicity of G, box(G) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes, i.e. each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let P = (S, P ) be a poset where S is the ground set and P is a reflexive, anti-symmetric and transitive binary relation on S. The dimension of P, dim(P) is the minimum integer t such that P can be expressed as the intersection of t total orders. Let GP be the underlying comparability graph of P, i.e. S is the vertex set and two vertices are adjacent if and only if they are comparable in P. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset P, box(GP )/(χ(GP ) − 1) ≤ dim(P) ≤ 2box(GP ), where χ(GP ) is the chromatic number of GP and χ(GP ) = 1. It immediately follows that if P is a height-2 poset, then box(GP ) ≤ dim(P) ≤ 2box(GP ) since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with the extended double cover of G, denoted as Gc: Note that Gc is a bipartite graph with partite sets A and B which are copies of V (G) such that corresponding to every u ∈ V (G), there are two vertices uA ∈ A and uB ∈ B and {uA, vB} is an edge in Gc if and only if either u = v or u is adjacent to v in G. Let Pc be the natural height-2 poset associated with Gc by making A the set of minimal elements and B the set of maximal elements. We show that box(G)These results have some immediate and significant consequences. The upper bound dim(P) ≤ 2box(GP ) allows us to derive hitherto unknown upper bounds for poset dimension such as dim(P) ≤ 2 tree-width (GP )+ 4, since boxicity of any graph is known to be at most its tree-width + 2. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree ∆ is O(∆ log 2 ∆) which is an improvement over the best known upper bound of ∆ 2 + 2. (2) There exist graphs with boxicity Ω(∆ log ∆). This disproves a conjecture that the boxicity of a graph is O(∆). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n 0.5−ǫ ) for any ǫ > 0, unless N P = ZP P .

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Section: Boxicitymentioning

confidence: 99%

“…Counter examples to the conjecture of [5]: Chandran et al [5] conjectured that boxicity of a graph is O(∆). We use a result by Erdős, Kierstead and Trotter [13] to show that there exist graphs with boxicity Ω(∆ log ∆), hence disproving the conjecture.…”

Section: Consequencesmentioning

confidence: 99%

Boxicity and Poset Dimension

Adiga

Bhowmick

Chandran

2010

Lecture Notes in Computer Science

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“…This was later strengthened by Yannakakis [6], and finally by Kratochvil [8] who showed that deciding whether boxicity of a graph is at most two itself is NP-complete. Recently Chandran et al [17] showed that for any graph G, box(G) ≤ χ(G 2 ) where G 2 is the square of graph G and χ is the chromatic number of a graph. From this they inferred that box(G) ≤ 2∆ 2 + 2, where ∆ is the maximum degree of G. Very recently this result was improved by Esperet [18], who showed that box(G) ≤ ∆ 2 + 2.…”

Section: Introductionmentioning

confidence: 99%

Boxicity of Circular Arc Graphs

Bhowmick

Chandran

2010

Graphs and Combinatorics

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A k-dimensional box is the cartesian product R 1 × R 2 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. Let G be a circular arc graph with maximum degree ∆. We show that if ∆ α but with ∆ = n (α−1) 2α + n 2α(α+1) + (α + 2). So the result cannot be improved substantially when α is large. Let r inf be minimum number of arcs passing through any point on the circle with respect to some circular arc representation of G. We also show that for any circular arc graph G, box(G) ≤ r inf + 1 and this bound is tight. Given a family of arcs F on the circle, the circular cover number L(F ) is the cardinality of the smallest subset F ′ of F such that the arcs in F ′ can cover the circle. Maximum circular cover number L max (G) is defined as the maximum value of L(F ) obtained over all possible family of arcs F that can represent G. We will show that if G is a circular arc graph with L max (G) > 4 then box(G) ≤ 3.

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“…More recent work has shown that graphs which are embeddable on a torus have boxicity at most 7, while graphs embeddable on a surface of genus g have boxicity at most 5g + 3 [28]. Boxicity has been bounded in terms of parameters such as treewidth [16] and maximum degree [1,13,27]. Boxicity of various other graph classes have also been studied [6,7,12,21], where in particular it was shown in [12] that there exist chordal graphs with arbitrarily high boxicity.…”

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confidence: 99%

Chronological Rectangle Digraphs

Huang

Manzer

2015

Electronic Notes in Discrete Mathematics

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Interval graphs admit elegant ordering and structural characterizations. A natural digraph analogue of interval graphs, called chronological interval digraphs, has recently been identified and studied.We introduce the class of chronological rectangle digraphs, and show that they are a higher dimensional analogue of chronological interval digraphs. A main goal of this thesis is to establish a foundation of knowledge about this class, including basic properties and an ordering characterization. Our most significant result is a forbidden induced subdigraph characterization for the series-parallel digraphs which are chronological rectangle. We also discuss obtaining chronological rectangle digraphs from orientations of graphs.In addition we introduce the related concept of the chronological interval dimension of a digraph, and determine the digraphs for which it is defined. Unit and proper chronological rectangle digraphs, defined analogously to unit and proper interval graphs, are also introduced and studied. iv

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Boxicity and maximum degree

Cited by 47 publications

References 8 publications

Boxicity and Poset Dimension

Boxicity and Poset Dimension

Boxicity of Circular Arc Graphs

Chronological Rectangle Digraphs

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