Intersection Graphs Of Segments Research Articles

Distributed Independent Sets in Interval and Segment Intersection Graphs

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

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.

Vertex Deletion into Bipartite Permutation Graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines ell _1 and ell _2, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is mathsf {NP}-complete by the classical result of Lewis and Yannakakis [20]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time {mathcal {O}}(9^k cdot n^9), and also give a polynomial-time 9-approximation algorithm.

{mathcal {U}}-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. https://doi.org/10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Box and Segment Intersection Graphs with Large Girth and Chromatic Number

We prove that there are intersection graphs of axis-aligned boxes in R3 and intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic number.

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.

Not all planar graphs are in PURE-4-DIR

We prove that some planar graphs are not intersection graphs of segments if only four slopes are allowed for the segments, and if parallel segments do not intersect. This refutes a conjecture of D. West [D. West, SIAM J. Discrete Math. Newsletter, 1991].

Optimality Program in Segment and String Graphs

Planar graphs are known to allow subexponential algorithms running in time 2^{O(sqrt{n})} or 2^{O(sqrt{n} log n)} for most of the paradigmatic problems, while the brute-force time 2^{varTheta (n)} is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in 2^{O(n^{2/3}log n)} by Fox and Pach (SODA’11), we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the Exponential Time Hypothesis (ETH). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time 2^{O(n^{2/3}log ^{O(1)}n)} on string graphs while an algorithm running in time 2^{o(n)} for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker lower bound, excluding a 2^{o(n^{2/3})} algorithm (under the ETH). The construction exploits the celebrated Erdős–Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set, but not to Min Dominating Set and Min Independent Dominating Set.

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.

On-Line Approach to Off-Line Coloring Problems on Graphs with Geometric Representations

The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that on-line graph coloring problems give rise to classes of game graphs with a natural geometric interpretation. We use this concept to estimate the chromatic number of graphs with geometric representations by finding, for appropriate simpler graphs, on-line coloring algorithms using few colors or proving that no such algorithms exist. We derive upper and lower bounds on the maximum chromatic number that rectangle overlap graphs, subtree overlap graphs, and interval filament graphs (all of which generalize interval overlap graphs) can have when their clique number is bounded. The bounds are absolute for interval filament graphs and asymptotic of the form $(\log\log n)^{f(\omega)}$ for rectangle and subtree overlap graphs, where $f(\omega)$ is a polynomial function of the clique number and $n$ is the number of vertices. In particular, we provide the first construction of geometric intersection graphs with bounded clique number and with chromatic number asymptotically greater than $\log\log n$. We also introduce a concept of $K_k$-free colorings and show that for some geometric representations, $K_3$-free chromatic number can be bounded in terms of clique number although the ordinary ($K_2$-free) chromatic number cannot. Such a result for segment intersection graphs would imply a well-known conjecture that $k$-quasi-planar geometric graphs have linearly many edges.

Refining the Hierarchies of Classes of Geometric Intersection Graphs

We analyse properties to show the strict containment between some natural classes of geometric intersection graphs. For example, we show that, for each integer k≥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, and the class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

Restricted Frame Graphs and a Conjecture of Scott

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.

Max point-tolerance graphs

A graph G is a max point-tolerance (MPT) graph if each vertex v of G can be mapped to a pointed-interval(Iv,pv) where Iv is an interval of R and pv∈Iv such that uv is an edge of G iff Iu∩Iv⊇{pu,pv}. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.

Finding cliques of maximum weight on a generalization of permutation graphs

We propose a dynamic programming procedure for computing the clique of maximum weight on a class of graphs arising in the solution of train timetabling problems. These graphs generalize, in two ways permutation graphs, defined as the intersection graphs of segments joining two parallel lines. First, two segments are joined by an edge not only if they intersect but also if their endpoints are sufficiently close. Second, two parallel segments may be joined by an edge even if they are arbitrarily far away from each other.

Triangle-Free Planar Graphs and Segment Intersection Graphs

Triangle-Free Planar Graphs and Segment Intersection Graphs

Intersection Graphs of Segments

Intersection graphs of segments (the class SEG) and of other simple geometric objects in the plane are considered. The results fall into two main areas: how difficult is the membership problem for a given class and how large are the pictures needed to draw the representations. Among others, we prove that the recognition of SEG-graphs is of the same complexity as the decision of solvability of a system of strict polynomial inequalities in the reals, i.e., as the decision of a special existentially quantified sentence in the theory of real closed fields, and thus it belongs to PSPACE. If the segments of the representation are restricted to lie in a fixed number ( k) of directions, we show that the corresponding recognition problem is NP-complete for every k ≥ 2. As for the size of representations, we show that the description of any segment representation, specifying the coordinates of the endpoints, may require exponential number of digits for certain n-vertex graphs. One of our main tools is a lemma, saying that given a representation R of a graph by segments, one may construct a larger graph whose each segment representation contains a homeomorphic copy of R.