Integer Representations of Convex Polygon Intersection Graphs

Abstract

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.

Homothetic triangle representations of planar graphs

“…This answers a conjecture of Lehmann that planar graphs are max-tolerance graphs (as max-tolerance graphs have shown to be exactly the intersection graphs of homothetic triangles [14]). Müller et al [17] proved that for some planar graphs, if the triangle corners have integer coordinates, then their intersection representation with homothetic triangles needs coordinates of order 2 Ω(n) , where n is the number of vertices. The following section is devoted to the proof of Theorem 2.…”