Comparability graphs and intersection graphs

Golumbic, Martin Charles; Rotem, Doron; Urrutia, Jorge

doi:10.1016/0012-365x(83)90019-5

Cited by 113 publications

(76 citation statements)

References 9 publications

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“…Box order is a particular type of geometrical containment order (See [16,28]). The result is as follows: the dimension of P is at most 2m if and only if it is a box order in m dimensions [18,20]. But note that boxicity is fundamentally different from box orders.…”

Section: Our Main Resultsmentioning

confidence: 99%

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: Our Main Resultsmentioning

confidence: 99%

Boxicity and Poset Dimension

Adiga

Bhowmick

Chandran

2010

Lecture Notes in Computer Science

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“…Proposition 2.1 ( [18], [33]) For each partial order ≺ on [n], there is a family of continuous functions …”

Section: Incomparability Graphs As String Graphsmentioning

confidence: 99%

“…In 1983, Golumbic, Rotem, and Urrutia [18] and Lovász [33] proved that every incomparability graph is a string graph (see Proposition 2.1). There are many string graphs, such as odd cycles of length at least five, which are not incomparability graphs.…”

Section: Introductionmentioning

confidence: 99%

“…To make our paper self-contained, in the next section we present the (few lines long) proof of the fact discovered by Golumbic, Rotem, and Urrutia [18] and Lovász [33] that every incomparability graph is a string graph. In Section 3, we establish a simple lemma showing that every dense graph has a cubic number of triangles K 3 or a quartic number of induced claws K 1,3 .…”

Section: Introductionmentioning

confidence: 99%

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String graphs and incomparability graphs

Fox

Pach

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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Dedicated to the memory of Paul Turán on his 100th birthday Given a collection C of curves in the plane, its string graph is defined as the graph with vertex set C, in which two curves in C are adjacent if and only if they intersect. Given a partially ordered set (P, <), its incomparability graph is the graph with vertex set P , in which two elements of P are adjacent if and only if they are incomparable.It is known that every incomparability graph is a string graph. For "dense" string graphs, we establish a partial converse of this statement. We prove that for every ε > 0 there exists δ > 0 with the property that if C is a collection of curves whose string graph has at least ε|C| 2 edges, then one can select a subcurve γ of each γ ∈ C such that the string graph of the collection {γ : γ ∈ C} has at least δ|C| 2 edges and is an incomparability graph. We also discuss applications of this result to extremal problems for string graphs and edge intersection patterns in topological graphs.

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“…Depending on whether the geometric objects under considerations are fixed in the plane (the "static" case) or can be moved in the plane through translation, rotation or even reflection (the "dynamic" case), different problems arise and have been studied. The majority of the investigations in both cases have focused on "simple" geometric figures such as rectangles [1,7,11,12,16], polygons [3,13,15,16], circles [2,14,15,16], angular regions [5,6,7,13,16], etc.…”

Section: Introductionmentioning

confidence: 99%

Geometric Containment and Partial Orders

Santoro

Sidney

et al. 1989

SIAM J. Discrete Math.

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Given two solid geometric figures on the plane (eg. rectangles) we say that A fits in B (denoted A < B) if there is a translation, a rotation and (if needed) a reflection that maps A into B. Given a family F of geometric figures, we say that the relation "<" on the elements of F is reducible to vector dominance if there exists an n and a mapping f:F AER n such that for A,BOEF A < B iff f(A) < f(B) coordinate by coordinate. A recent result states that if F is the set of all rectangles, "<" is not reducible to a vector dominance relation regardless of the finite value of n. In this paper we extend this result to other families of geometric figures and to a partial order obtained from quadratic polynomials.

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Comparability graphs and intersection graphs

Cited by 113 publications

References 9 publications

Boxicity and Poset Dimension

Boxicity and Poset Dimension

String graphs and incomparability graphs

Geometric Containment and Partial Orders

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