Boxicity and treewidth

Lecture Notes in Computer Science

2010

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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 .

Section: Boxicitymentioning

confidence: 99%

Boxicity and Poset Dimension

Adiga

Bhowmick

Lecture Notes in Computer Science

2010

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“…In a recent manuscript [7] the authors showed that for any graph G, box(G) ≤ tw(G)+2, where tw(G) is the treewidth of G. This result implies that the class of 'low boxicity' graphs properly contains the class of 'low treewidth graphs'. It is well known that almost all graphs on n vertices and m = cn edges (for a sufficiently large constant c) have Ω(n) treewidth [14].…”

Section: Introductionmentioning

confidence: 99%

“…Upper bounds for the boxicity of many other graph classes such as chordal graphs, AT-free graphs, permutation graphs etc. were shown in [7] by relating the boxicity of a graph with its treewidth. Researchers have also tried to generalize or extend the concept of boxicity in various ways.…”

Section: Introductionmentioning

confidence: 99%

Geometric Representation of Graphs in Low Dimension

Lecture Notes in Computer Science

Sivadasan

2006

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An axis-parallel k-dimensional box is a Cartesian product R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc.A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity 2 graphs has a 1 + 1 c log n approximation ratio for any constant c. 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.We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(∆ + 2) ln n⌉ dimension, where ∆ is the maximum degree of G. This algorithm implies that box(G) ≤ ⌈(∆ + 2) ln n⌉ for any graph G. Our bound is tight up to a factor of ln n.We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm.Though our general upper bound is in terms of maximum degree ∆, we show that for almost all graphs on n vertices, their boxicity is O(d av ln n) where d av is the average degree.

“…Remark on previous approach to boxicity of circular arc graphs: In [15] it has been shown that for any graph G, box(G) ≤ treewidth(G) + 2. If G is circular arc graph then treewidth(G) ≤ 2ω(G) − 1.…”

Section: Introductionmentioning

confidence: 99%

Boxicity of Circular Arc Graphs

Bhowmick

Graphs and Combinatorics

2010

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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.