Geometric Intersection Graphs Research Articles

Geometric random intersection graphs with general connection probabilities

Abstract Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$ . A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$ , where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$ . This gives rise to a random intersection graph on $\mathbb{R}^d$ . Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

Proper colorability of segment intersection graphs

We consider the vertex proper coloring problem for highly restricted instances of geometric intersection graphs of line segments embedded in the plane. Provided a graph in the class UNIT-PURE-k-DIR, corresponding to intersection graphs of unit length segments lying in at most k directions with all parallel segments disjoint, and provided explicit coordinates for segments whose intersections induce the graph, we show for k=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k = 4$$\\end{document} that it is NP-complete to decide if a proper 3-coloring exists, and moreover, #P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\#P$$\\end{document}-complete under many-one counting reductions to determine the number of such colorings. In addition, under the more relaxed constraint that segments have at most two distinct lengths, we show these same hardness results hold for finding and counting proper k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( k-1\\right) $$\\end{document}-colorings for every k≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ge 5$$\\end{document}. More generally, we establish that the problem of proper 3-coloring an arbitrary graph with m edges can be reduced in Om2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}\\left( m^2\\right) $$\\end{document} time to the problem of proper 3-coloring a UNIT-PURE-4-DIR graph. This can then be shown to imply that no 2on\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{o\\left( \\sqrt{n}\\right) }$$\\end{document} time algorithm can exist for proper 3-coloring PURE-4-DIR graphs under the Exponential Time Hypothesis (ETH), and by a slightly more elaborate construction, that no 2on\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{o\\left( \\sqrt{n}\\right) }$$\\end{document} time algorithm can exist for counting the such colorings under the Counting Exponential Time Hypothesis (#ETH). Finally, we prove an NP-hardness result for the optimization problem of finding a maximum order proper 3-colorable induced subgraph of a UNIT-PURE-4-DIR graph.

Extending Partial Representations of Circle Graphs in Near-Linear Time

AbstractThe partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph $$H \subseteq G$$ H ⊆ G and a representation $$\mathcal R'$$ R ′ of H. The question is whether G admits a representation $$\mathcal R$$ R whose restriction to H is $$\mathcal R'$$ R ′ . We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in $$O((n + m) \alpha (n + m))$$ O ( ( n + m ) α ( n + m ) ) time, thereby improving over an $$O(n^3)$$ O ( n 3 ) -time algorithm by Chaplick et al. (J Graph Theory 91(4), 365–394, 2019). The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest.

Recognizing geometric intersection graphs stabbed by a line

In this paper, we determine the computational complexity of recognizing two graph classes, groundedL-graphs and stabbable grid intersection graphs. An L-shape is made by joining the bottom end-point of a vertical (|) segment to the left end-point of a horizontal (−) segment. The top end-point of the vertical segment is known as the anchor of the L-shape. Grounded L-graphs are the intersection graphs of L-shapes such that all the L-shapes' anchors lie on the same horizontal line. We show that recognizing grounded L-graphs is NP-complete. This answers an open question asked by Jelínek & Töpfer (Electron. J. Comb., 2019).Grid intersection graphs are the intersection graphs of axis-parallel line segments in which two vertical (similarly, two horizontal) segments cannot intersect. We say that a (not necessarily axis-parallel) straight line ℓ stabs a segment s, if s intersects ℓ. A graph G is a stabbable grid intersection graph (▪) if there is a grid intersection representation of G in which the same line stabs all its segments. We show that recognizing ▪ graphs is NP-complete, even on a restricted class of graphs. This answers an open question asked by Chaplick et al. (Order, 2018).

A Survey on Variant Domination Problems in Geometric Intersection Graphs

Given a graph [Formula: see text] with vertex set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex is either in [Formula: see text] or is adjacent to a vertex in [Formula: see text]. There are a lot of variants for dominating sets, such as connected dominating sets, total dominating sets, total restricted dominating sets, secure (connected, total) dominating sets, etc. Geometric intersection graphs are graphs for which there is a bijection [Formula: see text] between the vertices and a set [Formula: see text] of geometric objects (for example, disks, rectangles, etc.) such that there is an edge between two vertices [Formula: see text] and [Formula: see text] if and only if the objects [Formula: see text] and [Formula: see text] intersect. In this paper, we offer a survey about complexity and algorithmic results on variant domination problems in geometric intersection graphs.

Maximum Bipartite Subgraphs of Geometric Intersection Graphs

We study the Maximum Bipartite Subgraph ([Formula: see text]) problem, which is defined as follows. Given a set [Formula: see text] of [Formula: see text] geometric objects in the plane, we want to compute a maximum-size subset [Formula: see text] such that the intersection graph of the objects in [Formula: see text] is bipartite. We first give an [Formula: see text]-time algorithm that computes an almost optimal solution for the problem on circular-arc graphs. We show that the [Formula: see text] problem is [Formula: see text]-hard on geometric graphs for which the maximum independent set is [Formula: see text]-hard (hence, it is [Formula: see text]-hard even on unit squares and unit disks). On the other hand, we give a [Formula: see text] for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks, and unit-height axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph ([Formula: see text]) problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the [Formula: see text] except the intersection graph induced by the set [Formula: see text] needs to be triangle-free only (instead of being bipartite).

Grounded L-Graphs Are Polynomially chi -Bounded

A grounded L-graph is the intersection graph of a collection of “L” shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number omega has chromatic number at most 17omega ^4. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially chi -bounded. We also survey chi -boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.

Maximum Matchings in Geometric Intersection Graphs

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in Ohspace{0.33325pt}(rho ^{3omega /2}n^{omega /2}) time with high probability, where rho is the density of the geometric objects and omega >2 is a constant such that ntimes n matrices can be multiplied in O(n^omega ) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{omega /2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Psi ] can be found in Ohspace{0.33325pt}(Psi ^6log ^{11}hspace{-0.55542pt}n + Psi ^{12 omega } n^{omega /2}) time with high probability.

Algorithmic aspects of secure domination in unit disk graphs

Given a graph G with vertex set V, a set S⊆V is a secure dominating set of G if S is a dominating set of G and if for every vertex u∈V∖S, there exists a vertex v∈S adjacent to u such that (S∪{u})∖{v} is a dominating set of G.The minimum secure dominating set (or, for short, MSDS) problem asks to find an MSDS in a given graph. In this paper, we first show that the decision version of the MSDS problem is NP-complete in unit disk graphs, even in grid graphs. Secondly, we give an O(n+m) time t-approximation algorithm for the MSDS problem in several geometric intersection graphs which are K1,t-free for some integer t≥3. Finally, we propose a PTAS for the MSDS problem in unit disk graphs.

Finding geometric representations of apex graphs is NP-hard

Planar graphs can be represented as the intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., SODA 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is NP-hard as well.More precisely, we show that for every fixed positive integer g and every graph class G such that ▪, it is NP-hard to decide whether an input graph belongs to the graph class G, even when the inputs are restricted to apex graphs of girth g. Here, ▪ is the class of intersection graphs of axis-parallel line segments (where horizontal segments intersect only vertical segments), and ▪ is the class of intersection graphs of simple curves (where two intersecting curves cross each other exactly once) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (COCOON, 2007).Most known reductions for earlier proofs of NP-hardness for these problems are from variants of 3-SAT (mainly Planar 3-Connected 3-SAT). We reduce from the ▪▪▪ problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric intersection graphs.

Space-efficient algorithms for reachability in directed geometric graphs

The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem.For intersection graphs of Jordan regions, we show how to obtain a “good” vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and O(m1/2log⁡n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial time and O(m1/2log⁡n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ϵ>0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n1/4+ϵ) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.

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.

The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

We define a class of problems whose input is an n-sized set of d-dimensional vectors, and where the problem is first-order definable using comparisons between coordinates. This class captures a wide variety of tasks, such as complex types of orthogonal range search, model-checking first-order properties on geometric intersection graphs, and elementary questions on multidimensional data like verifying Pareto optimality of a choice of data points. Focusing on constant dimension d, we show that any such k-quantifier, d-dimensional problem is solvable in $$O(n^{k-1} \log ^{d-1} n)$$ time. Furthermore, this algorithm is conditionally tight up to subpolynomial factors: we show that assuming the 3-uniform hyperclique hypothesis, there is a k-quantifier, $$(3k-3)$$ -dimensional problem in this class that requires time $$\Omega (n^{k-1-o(1)})$$ . Towards identifying a single representative problem for this class, we study the existence of complete problems for the 3-quantifier setting (since 2-quantifier problems can already be solved in near-linear time $$O(n\log ^{d-1} n)$$ , and k-quantifier problems with $$k>3$$ reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination $$\mathsf {VCND}_d$$ (Given three sets of vectors X, Y and Z of dimension d, d and 2d, respectively, is there an $$x \in X$$ and a $$y \in Y$$ so that their concatenation $$x \circ y$$ is not dominated by any $$z \in Z$$ , where vector u is dominated by vector v if $$u_i \le v_i$$ for each coordinate $$1 \le i \le d$$ ), and determine it as the “unique” candidate to be complete for this class (under fine-grained assumptions).

The balanced connected subgraph problem for geometric intersection graphs

We study the ▪ (shortly, ▪) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. We are given a ▪ graph G=(V,E) and an integer k. Each vertex in V is colored with either “▪” or “▪”. The BCS problem seeks an induced connected subgraph H of size at least k in G such that H is ▪, i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On the one hand, we prove that the BCS problem is NP-complete on the unit-disk, outer-string, grid, and unit-square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on the interval, circular-arc, and permutation graphs. Our algorithm for interval and circular-arc graphs solves the more general problem of computing a minimum cardinality ▪ in the same classes of graphs, which may be of independent interest.

Contraction Bidimensionality of Geometric Intersection Graphs

Given a graph G, we define \(\mathbf{bcg}(G)\) as the minimum k for which G can be contracted to the uniformly triangulated grid \(\Gamma _{k}\). A graph class \({\mathcal {G}}\) has the SQGC property if every graph \(G\in {\mathcal {G}}\) has treewidth \(\mathcal {O}(\mathbf{bcg}(G)^{c})\) for some \(1\le c<2\). The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQGC property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.

The Balanced Connected Subgraph Problem for Geometric Intersection Graphs

The Balanced Connected Subgraph Problem for Geometric Intersection Graphs

On H-Topological Intersection Graphs

Biró et al. (Discrete. Math 100(1–3):267–279, 1992) introduced the concept of H-graphs, intersection graphs of connected subgraphs of a subdivision of a graph H. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying several classical computational problems on H-graphs: recognition, graph isomorphism, dominating set, clique, and colorability. We negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is \(\textsf {NP}\)-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T-graphs, for each fixed tree T. For the special case when T is a star \(S_d\) of degree d, we have an \(\mathcal{O}(n^{3.5})\)-time algorithm. We give \(\textsf {FPT}\)- and \(\textsf {XP}\)-time algorithms solving the minimum dominating set problem on \(S_d\)-graphs and H-graphs, parametrized by d and the size of H, respectively. The algorithm for H-graphs adapts to an \(\textsf {XP}\)-time algorithm for the independent set and the independent dominating set problems on H-graphs. If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph has a Helly H-representation, the clique problem is polynomial-time solvable. Further, we show that both the k-clique and the list k-coloring problems are solvable in FPT-time on H-graphs, parameterized by k and the treewidth of H. In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that H-graphs have at most \(n^{O(\Vert H\Vert )}\) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to \(O(\Vert H\Vert n^2)\).

Computing inductive vertex orderings

Inductive vertex orderings or edge orientations of graphs, such as inductive k-independence and k-simpliciality orderings, and k-perfect orientation, arise naturally in geometric intersection graphs and have been systematically studied in Ye and Borodin (2012) [4] and Kammer and Tholey (2014) [5]. In general, those orderings and orientations are hard to compute, even approximately. In this note, we show that an inductive k-independence ordering can be computed in k-simplicial graphs, and a k-approximation (that is, a solution with approximation ratio at most k) to the optimal inductive independence ordering can be computed in k-perfectly orientable graphs. As a result, we obtain a k-approximation (2k-approximation) algorithm for maximum weight independent sets in k-simplicial (k-perfectly orientable, respectively) graphs that runs in polynomial time independent of k without prior knowledge of the ordering.

Minimum cuts in geometric intersection graphs

Let D be a set of n disks in the plane. The disk graphGD for D is the undirected graph with vertex set D in which two disks are joined by an edge if and only if they intersect. The directed transmission graphGD→ for D is the directed graph with vertex set D in which there is an edge from a disk D1∈D to a disk D2∈D if and only if D1 contains the center of D2.Given D and two non-intersecting disks s,t∈D, we show that a minimum s-t vertex cut in GD or in GD→ can be found in O(n3/2polylogn) expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks.As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip S bounded by two vertical lines, Lℓ and Lr, and a collection D of disks. Let a be a point in S above all disks of D, and let b a point in S below all disks of D. The task is to find a curve from a to b that lies in S and that intersects as few disks of D as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in O(n3/2polylogn) expected time.

A Framework for Exponential-Time-Hypothesis--Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs ...