Convex Polygon Intersection Graphs

Leeuwen, Erik Jan van; Leeuwen, Jan van

doi:10.1007/978-3-642-18469-7_35

Cited by 8 publications

(11 citation statements)

References 19 publications

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“…It follows that trans(P ) and hom(P ) have small polynomial-size certificates, hence recognition problems in NP, for all convex polygons P with rational corner points. This improves on earlier results about the recognition of these graphs [24,31]. Downloaded 01/31/14 to 131.211.105.115.…”

Section: Our Resultssupporting

confidence: 87%

“…Part (i) of Theorem 1 is a considerable improvement over a result of Czyzowicz et al [7], who only showed that t U (n) ≤ 2 n−1 , where U denotes the unit square. Theorem 2 greatly improves a recent bound in [31], where it was shown that h P (n) = 2 O(n 4 ) for convex polygons with rational corner points.…”

Section: Our Resultssupporting

confidence: 71%

“…A number of them relate to the complexity of recognizing trans(P ) and hom(P ) graphs. As in [31], our results imply that both recognition problems are in NP, on the assumption that P has rational corner points. This, however, does not settle the problem for polygons P that do not have rational corner points.…”

Section: Conclusion and Further Worksupporting

confidence: 70%

“…This implies that w corresponds to a homothets, resp., translates, representation of G with all corner points on integer points, and every coordinate of every corner point is O(C n ) = 2 O(n) . Theorem 4 considerably improves the bound given in [31]. We also note that Lemma 3 and Theorem 4 and their proofs generalize to higher dimensions.…”

Section: Upper Bound For Translates and Homothetssupporting

confidence: 52%

“…In this case it can be shown that for every G ∈ trans(P ) there always is a collection A = {A(v) : v ∈ V (G)} where each set is of the form A(v) = p(v) + λP with λ ≥ 0, such that A realizes G and all corners of A(v) lie on Z 2 . (See, e.g., [31] or the proof of Theorem 2 below.) We call such an A an integer translate realization of G.…”

Section: Problem Descriptionmentioning

confidence: 99%

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Integer Representations of Convex Polygon Intersection Graphs

Müller¹,

Leeuwen²,

Leeuwen³

2013

SIAM J. Discrete Math.

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Abstract. We determine tight bounds on the smallest-size integer grid needed to represent the n-node intersection graphs of a convex polygon P with P given in rational coordinates. The intersection graphs use only polygons that are geometrically similar to P (translates or homothets) and must be represented such that each corner of each polygon lies on a point of the grid. We show the following generic results: if P is a parallelogram and only translates of P are used, then an Ω(n 2 ) × Ω(n 2 ) grid is sufficient and is needed for some graphs; if P is any other convex polygon and only translates of P are used, then a 2 Ω(n) × 2 Ω(n) grid is sufficient and is needed for some graphs; if P is any convex polygon and arbitrary homothets of P are allowed, then a 2 Ω(n) × 2 Ω(n) grid is sufficient and is needed for some graphs. The results substantially improve earlier bounds and settle the complexity of representing convex polygon intersection graphs. The results also imply small polynomial certificates for the recognition problem for all graph classes considered. 1. Introduction. Intersection graphs are widely used for modeling purposes [10]. Many variants of these graphs have been studied, from a graph-theoretic and a computational perspective (see, e.g., [20,29]). In this paper we consider convex polygon intersection graphs, which arise naturally in planning and layout problems in the plane. We analyze the representation problem for these graphs and determine the complexity of representing them as geometric structures in computer memory. In particular, we will prove tight upper and lower bounds on the value of N = 2 b(n) such that any n-node convex polygon intersection graph can be realized with each corner of each polygon lying on a point of the N × N grid. The precise definitions will be given below. We will see that the bounds depend nontrivially on the underlying base polygon of the graphs.We first define intersection graphs more precisely. Let A = {A i : i ∈ I} be any collection of sets or objects. The intersection graph of A is a graph G = (I, E) with vertex set I and edge ij ∈ E if and only if A i ∩ A j = ∅. If A has G as its intersection graph, then we say that A realizes G. By constraining the allowed collections A one obtains different classes of intersection graphs. For each class the following questions arise [29]: the recognition problem ("determine whether a given graph is realized as a member of the given class") and the representation problem ("represent a graph as an intersection graph of a given class"). Given an n-node intersection graph G, we typically want a realization of G by an allowed n-element collection A that gives a representation in a minimum amount of memory space.

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Section: Our Resultssupporting

confidence: 87%

Section: Our Resultssupporting

confidence: 71%

Section: Conclusion and Further Worksupporting

confidence: 70%

Section: Upper Bound For Translates and Homothetssupporting

confidence: 52%

Section: Problem Descriptionmentioning

confidence: 99%

See 3 more Smart Citations

Integer Representations of Convex Polygon Intersection Graphs

Müller¹,

Leeuwen²,

Leeuwen³

2013

SIAM J. Discrete Math.

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Representing Graphs and Hypergraphs by Touching Polygons in 3D

Evans

Rzążewski

Saeedi

et al. 2019

Lecture Notes in Computer Science

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Contact representations of graphs have a long history. Most research has focused on problems in 2d, but 3d contact representations have also been investigated, mostly concerning fully-dimensional geometric objects such as spheres or cubes. In this paper we study contact representations with convex polygons in 3d. We show that every graph admits such a representation. Since our representations use super-polynomial coordinates, we also construct representations on grids of polynomial size for specific graph classes (bipartite, subcubic). For hypergraphs, we represent their duals, that is, each vertex is represented by a point and each edge by a polygon. We show that even regular and quite small hypergraphs do not admit such representations. On the other hand, the two smallest Steiner triple systems can be represented.

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