Intersection Graphs Of Disks Research Articles

Counting disk graphs

Abstract A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of same radius disks in the plane. We give upper and lower bounds on the number of labelled unit disk and disk graphs on n vertices. We show that the number of disk graphs on n vertices is n ( 3 + o ( 1 ) ) n and the number of unit disk graphs on n vertices is n ( 2 + o ( 1 ) ) n .

MAXIMUM AREA INDEPENDENT SETS IN DISK INTERSECTION GRAPHS

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

The Clique Problem in Intersection Graphs of Ellipses and Triangles

Intersection graphs of disks and of line segments, respectively, have been well studied, because of both practical applications and theoretically interesting properties of these graphs. Despite partial results, the complexity status of the Clique problem for these two graph classes is still open. Here, we consider the Clique problem for intersection graphs of ellipses, which, in a sense, interpolate between disks and line segments, and show that the problem is APX-hard in that case. Moreover, this holds even if for all ellipses, the ratio of the larger over the smaller radius is some prescribed number. Furthermore, the reduction immediately carries over to intersection graphs of triangles. To our knowledge, this is the first hardness result for the Clique problem in intersection graphs of convex objects with finite description complexity. We also describe a simple approximation algorithm for the case of ellipses for which the ratio of radii is bounded.

Well-Separated Pair Decomposition for the Unit-Disk Graph Metric and Its Applications

We extend the classic notion of well-separated pair decomposition [P. B. Callahan and S. R. Kosaraju, J. ACM, 42 (1975), pp. 67--90] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant $c\geq 1$, there exists a c-well-separated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where $k\geq 3$, there exists a c-well-separated pair decomposition with O(n2-2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric.

Representing graphs by disks and balls (a survey of recognition-complexity results)

Practical applications, like radio frequency assignments, led to the definition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper, we survey recognition-complexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about bounded-ratio disk representations. We prove that the recognition of disk-intersection graphs (in the unbounded ratio case) is NP-hard. This result is proved in a more general setting of noncrossing arc-connected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unit-ball contact graphs is NP-hard in dimensions 3,4, and 8 (24).

On Coloring Unit Disk Graphs

In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal-sized disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving on a result of Clark et al. [2] it is shown that the coloring problem for UD graphs remains NP-complete for any fixed number of colors k≥ 3 . Furthermore, a new 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques.

Testing the necklace condition for shortest tours and optimal factors in the plane

A tour T of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in T . It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour. In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O( n 2 log n) time. If a tour is given, then we can test in O( n 2) time whether or not this tour is a necklace-tour. Both algorithms can be generalized to ƒ-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.

A study on two geometric location problems

Given a convex n-gon, we can draw n disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the n disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and Pérez-Lantero (Discrete Mathematics, 2020), proved to be planar and Hamiltonian. In this paper, we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an O(n)-time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in O(n) time. We further study the independence number, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex n-gon we can select at least ⌈n/4⌉ pairwise disjoint disks, and we prove that for every n≥3 there exist convex n-gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs.