Intersection Graphs Of Disks Research Articles

Online coloring of disk graphs

In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online L(2,1)-labeling.

Dynamic Connectivity in Disk Graphs

Let S subseteq mathbb {R}^2 be a set of nsites in the plane, so that every site s in S has an associated radiusr_s > 0. Let mathcal {D}(S) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites s, t in S if and only if the disks with centers s, t and radii r_s, r_t intersect. Our goal is to design data structures that maintain the connectivity structure of mathcal {D}(S) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix r_s = 1, for all sites s in S. For this case, we describe a data structure that has O(log ^2 n) amortized update time and O(log n/log log n) query time. Second, we look at disk graphs with bounded radius ratioPsi , i.e., for all s in S, we have 1 le r_s le Psi , for a parameter Psi that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time O(Psi log ^{4} n) and query time O(log n/log log n). This improves the currently best update time by a factor of Psi . In the incremental case, we achieve logarithmic dependency on Psi , with a data structure that has O(alpha (n)) amortized query time and O(log Psi log ^{4} n) amortized expected update time, where alpha (n) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of O(log n/log log n) that supports deletions in O(nlog Psi log ^{4} n) overall expected time for disk graphs with bounded radius ratio Psi and O(nlog ^{5} n) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures.

On Embeddability of Unit Disk Graphs Onto Straight Lines

Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane. Deciding whether there exists an embedding of a given unit disk graph, i.e. unit disk graph recognition, is an important geometric problem, and has many application areas. In general, this problem is known to be \(\exists \mathbb {R}\)-complete. In some applications, the objects that correspond to unit disks, have predefined (geometrical) structures to be placed on. Hence, many researchers attacked this problem by restricting the domain of the disk centers. Following the same line, we also describe a polynomial-time reduction which shows that deciding whether a graph can be realized as unit disks onto given straight lines is NP-hard, when the given lines are parallel to either x-axis or y-axis. Adjusting the reduction, we also show that this problem is NP-complete when the given lines are only parallel to x-axis. We obtain those results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.

An algorithmic framework for the single source shortest path problem with applications to disk graphs

Shortest path problems are among the fundamental problems in graph theory. It is folklore that the unweighted single source shortest path (SSSP) problem in general graphs can be solved optimally with breadth first search (BFS) in O(n+m) time. In this paper, we develop an algorithmic framework that generalizes a batched BFS approach to give efficient SSSP algorithms for several graph classes. The running time of these algorithms depends on the running time of three main ingredients. The first is a preprocessing step, to define a shortcut graph that maintains some distance information. Then during one run of the algorithm repeatably there are the steps of efficiently finding a set of candidate vertices adjacent in the shortcut graph to a given set of vertices and finally finding the subset of the candidate vertices that actually form an edge in the original graph.A disk graphD(S) is a graph that is defined on a set S of point sites in R2, where each site s∈S has an associated radius rs. The vertex set of D(S) is S and two sites s,t are connected by an edge st in D(S) if and only if the disks induced by s and t intersect. These graphs are also called the intersection graph of disks. Our results are algorithms that use the framework to efficiently solve the SSSP problem in intersection graphs. For disk graphs in the L2-metric, we can show that after O(nlog2⁡n) preprocessing time we can solve the SSSP problem in O(nlog⁡n) time. This significantly improves the previous best bound of O(nlog4⁡n)[1,2]. In the case of intersection graphs of axis-parallel squares, we are even able to reduce the preprocessing time to an optimal O(nlog⁡n). As intersection graphs of axis parallel squares are equivalent to disk graphs in the L1- and L∞-metric the result carries over to these metrics.To show further applications of our framework, we restate the classical BFS, and also the optimal SSSP algorithm for unit disk graphs by Chan and Skrepetos [3] in our framework, showing its robustness.

Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let mathcal {G}^{times }(F) be its intersection graph. A balanced clique-based separator of mathcal {G}^{times }(F) is a set mathcal {mathcal {S}} consisting of cliques whose removal partitions mathcal {G}^{times }(F) into components of size at most delta n, for some fixed constant delta <1. The weight of a clique-based separator is defined as sum _{Cin mathcal {mathcal {S}}}log (|C|+1). Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then mathcal {G}^{times }(F) admits a balanced clique-based separator of weight O(sqrt{n}). We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight O(sqrt{n}), which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3}log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(sqrt{n}log n). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3}log n). (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(sqrt{n}+rlog (n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover), for Feedback Vertex Set, and for q-Coloring for constant q in these graph classes.

Nearly Optimal Planar $k$ Nearest Neighbors Queries under General Distance Functions

We study the $k$ nearest neighbors problem in the plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals under a general family of distance functions including the $L_p$ norms and additively weighted Euclidean distances. We compose a static data structure for this general setting with nearly optimal $O(n\log\log n)$ space, optimal $O(\log n+k)$ query time, and optimal expected $O(n\log n)$ preprocessing time. We also devise a dynamic data structure (that allows insertions and deletions of sites) with $O(n\log n)$ space, $O(\log^2n+k\log n)$ query time, and expected amortized $O(\log^2 n)$ insertion time and $O(\log^4n)$ deletion time, matching the best known time complexities for point sites in the Euclidean metric and improving many applications such as dynamic minimum spanning trees in a general planar metric and dynamic connectivity in disk intersection graphs. Our results, to some extent, indicate that for the $k$ nearest neighbors problem, general distance functions may share the same time complexities with point sites in the Euclidean metric. To achieve this progress, we design vertical shallow cuttings of linear size for general distance functions. Vertical shallow cuttings are a key technique to tackle the $k$ nearest neighbors problem for point sites in the Euclidean metric, while existing generalizations to general distance functions are either not vertical or not of linear size. Our innovation is a new random sampling technique for the analysis of geometric structures, and since this technique provides a new way to develop geometric algorithms, we believe it is of independent interest.

On the intersection graph of the disks with diameters the sides of a convex [formula omitted]-gon

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.

On the speed of algebraically defined graph classes

The speed of a class of graphs counts the number of graphs on the vertex set {1,…,n} inside the class as a function of n. In this paper, we investigate this function for many classes of graphs that naturally arise in discrete geometry, for example intersection graphs of segments or disks in the plane. While upper bounds follow from Warren's theorem (a variant of a theorem of Milnor and Thom), all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving an essentially tight lower bound for the number of graphs on {1,…,n} whose edges are defined using the signs of a given finite list of polynomials, assuming these polynomials satisfy some reasonable conditions. This in particular implies lower bounds for the speed of many different classes of intersection graphs, which essentially match the known upper bounds. Our general result also gives essentially tight lower bounds for counting containment orders of various families of geometric objects, including circle orders and angle orders. Some of the applications presented in this paper are new, whereas others recover results of Alon-Scheinerman, Fox, McDiarmid-Müller and Shi. For the proof of our result we use some tools from algebraic geometry and differential topology.

Dynamic Planar Voronoi Diagrams for General Distance Functions and Their Algorithmic Applications

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include L_p-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses O(n log ^3 n) storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat, and Sharir which required O(n^{varepsilon }) time for an update and O(log n) time for a query [SICOMP 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an O(n^{varepsilon }) factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques from the literature. Along the way, we obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for computing the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest-neighbor context, under both types of norm). To analyze this algorithm, we also improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we combine our vertical shallow cutting construction with Chan’s algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in {{mathbb {R}}}^3. Along the way, we also revisit Chan’s technique and present a variant that uses a single binary counter, with a simpler analysis and improved amortized deletion time (by a logarithmic factor; the insertion and query costs remain asymptotically the same).

The intersection graph of the disks with diameters the sides of a convex [formula omitted]-gon

Given a convex polygon of n sides, one 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. We introduce the study of this graph by proving that it is planar for every convex polygon.

Formula omitted]-labeling of disk intersection graphs

In this paper we study the problem of L(2,1)-labeling of intersection graphs of disks. An L(2,1)-labeling is a mapping from the vertex set of the graph to non-negative integers, in which labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices at distance 2 are different. The span of an L(2,1)-labeling is the difference between the maximum and the minimum label used, and the span λ(G) of a graph G is the minimum span of an L(2,1)-labeling of G.We show that if G is an intersection graph of disks, then λ(G)≤45Δ2+25Δ+20. Moreover, if all disks in the geometric representation of G have equal radii, then λ(G)≤14Δ+11. We also present a refined upper bound for unit disk intersection graphs with small maximum degree.

Balanced line separators of unit disk graphs

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O((m+n)log⁡n) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting O(m+n) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m=0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(m)).

On Forbidden Induced Subgraphs for Unit Disk Graphs

A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non-unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non-unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a \(C_4\)-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation.

Online Coloring and $L(2,1)$-Labeling of Unit Disk Intersection Graphs

In this paper we give a family of online algorithms for the classical coloring and the $L(2,1)$-labeling problems of unit disk intersection graphs. In the $L(2,1)$-labeling we ask for an assignment of nonnegative integers to the vertices of the input graph, such that adjacent vertices get labels that differ by at least 2, and vertices with a common neighbor get different labels. In particular, we present a coloring algorithm with competitive ratio less than 5, which makes it the currently best online coloring algorithm for unit disk intersection graphs. Our algorithms make use of a geometric representation of such graphs and are inspired by previous results but have better competitive ratios. The improvement comes from a novel application of a fractional and a $b$-fold coloring of the plane, which is in turn a variation of the Hadwiger--Nelson problem. Our method can also be adapted successfully for other classes of geometric intersection graphs.

Co-Bipartite Neighborhood Edge Elimination Orderings

In SODA 2001, Raghavan and Spinrad introduced robust algorithms as a way to solve hard combinatorial graph problems in polynomial time even when the input graph falls slightly outside a graph class for which a polynomial-time algorithm exists. As a leading example, the Maximum Clique problem on unit disk graphs (intersection graphs of unit disks in the plane) was shown to have a robust, polynomial-time algorithm by proving that such graphs admit a co-bipartite neighborhood edge elimination ordering (CNEEO). This begs the question whether other graph classes also admit a CNEEO.In this paper, we answer this question positively, and identify many graph classes that admit a CNEEO, including several graph classes for which no polynomial-time recognition algorithm exists (unless P=NP). As a consequence, we obtain robust, polynomial-time algorithms for Maximum Clique on all identified graph classes.We also prove some negative results, and identify graph classes that do not admit a CNEEO. This implies an almost-perfect dichotomy for subclasses of perfect graphs.

Refining the Hierarchies of Classes of Geometric Intersection Graphs

We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties:A graph $G$ is outerplanar if and only if the 1-subdivision of $G$ is outer-segment.For each integer $k\ge 1$, the class of intersection graphs of segments with $k$ different lengths is a strict subclass of the class of intersection graphs of segments with $k+1$ different lengths.For each integer $k\ge 1$, the class of intersection graphs of disks with $k$ different sizes is a strict subclass of the class of intersection graphs of disks with $k+1$ different sizes.The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

Shortest paths in intersection graphs of unit disks

Let G be a unit disk graph in the plane defined by n disks whose positions are known. For the case when G is unweighted, we give a simple algorithm to compute a shortest path tree from a given source in O(nlog⁡n) time. For the case when G is weighted, we show that a shortest path tree from a given source can be computed in O(n1+ε) time, improving the previous best time bound of O(n4/3+ε).

In-place algorithms for computing a largest clique in geometric intersection graphs

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R2. First, we propose an O(n2logn) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangles, the time complexity can be slightly improved to O(nlogn+nK), where K is the size of the maximum clique. For disk graphs, we consider two variations of the maximum clique problem, namely geometric clique and graphical clique. The time complexity of our algorithm for finding the largest geometric clique is O(mlogn+n2) where m is the number of edges in the disk graph, and it works for disks of arbitrary radii. For graphical clique, our proposed algorithm works for unit disks (i.e., of same radii) and the worst case time complexity is O(n2+m(n+K3)); m is the number of edges in the unit disk intersection graph and K is the size of the largest clique in that graph. It uses O(n3) time in-place computation of maximum matching in a bipartite graph, where the vertices are given in an array, and the existence of an edge between a pair of vertices can be checked by an oracle on demand (from problem specification) in O(1) time. This problem is of independent interest. All these algorithms need O(1) work space in addition to the input array.

The number of disk graphs

A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labeled unit disk and disk graphs on n vertices. We show that the number of unit disk graphs on n vertices is n2n⋅α(n)n and the number of disk graphs on n vertices is n3n⋅β(n)n, where α(n) and β(n) are Θ(1). We conjecture that there exist constants α,β such that the number of unit disk graphs is n2n⋅(α+o(1))n and the number of disk graphs is n3n⋅(β+o(1))n.

Integer realizations of disk and segment graphs

A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. Every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the integer grid and equal (integer) radius; and every segment graph can be realized by segments whose endpoints lie on the integer grid. Here we show that there exist disk graphs on n vertices such that in every realization by integer disks at least one coordinate or radius is 22Ω(n) and on the other hand every disk graph can be realized by disks with integer coordinates and radii that are at most 22O(n); and we show the analogous results for unit disk graphs and segment graphs. For (unit) disk graphs this answers a question of Spinrad, and for segment graphs this improves over a previous result by Kratochvíl and Matoušek.