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On the stab number of rectangle intersection graphs [article]

Dibyayan Chakraborty, Mathew C. Francis
2018 arXiv   pre-print
Tight upper bounds on the exact stab number of split graphs with boxicity at most 2 and block graphs are also given.  ...  We introduce the notion of stab number and exact stab number of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2.  ...  In particular, we show (a) that all rectangle intersection graphs that are also split graphs have exact stab number at most 3 and that this bound is tight, and (b) an upper bound of log m on the exact  ...

Development of Algorithm for Identification of Area for Maximum Coverage and Interference

Janak Gupta, Pankaj Kumar
2017 International Journal of Computer Applications
For a point set P, where no two points have the same x or y coordinates, derive an upper bound on the size of the stabbing set of axis-parallel rectangles induced by each pair of points a,b ∈ P as the  ...  For a point set P in convex position, derive a lower bound on the size of the stabbing set axis-parallel rectangles induced by each pair of points a,b∈P as the diagonal of the rectangles.  ...  So we have seen that in the case when two rectangles corresponding to adjacent edge intersect there does not exist third rectangle which can intersect with both of them.  ...

Fixed-parameter algorithms for Cochromatic Number and Disjoint Rectangle Stabbing via iterative localization

Pinar Heggernes, Dieter Kratsch, Daniel Lokshtanov, Venkatesh Raman, Saket Saurabh
2013 Information and Computation
This NP-complete problem is equivalent to deciding whether the cochromatic number of a given permutation graph on n vertices is at most k.  ...  Our algorithm solves in fact a more general problem: within the mentioned running time, it decides whether the cochromatic number of a given perfect graph on n vertices is at most k.  ...  In particular, our algorithm partitions the n rectangles into two groups R 1 and R 2 with at most n/2 rectangles in each group and runs recursively on the two groups.  ...

Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing [chapter]

Pinar Heggernes, Dieter Kratsch, Daniel Lokshtanov, Venkatesh Raman, Saket Saurabh
2010 Lecture Notes in Computer Science
In fact, we give a more general result: within the mentioned running time, one can decide whether the cochromatic number of a given perfect graph on n vertices is at most k.  ...  This NP-complete problem is equivalent to deciding whether the cochromatic number, partitioning into the minimum number of cliques or independent sets, of a given permutation graph on n vertices is at  ...  In particular, our algorithm partitions the n rectangles into two groups R 1 and R 2 with at most n/2 rectangles in each group and runs recursively on the two groups.  ...

Greedy is good: An experimental study on minimum clique cover and maximum independent set problems for randomly generated rectangles [article]

Ritankar Mandal and Anirban Ghosh and Sasanka Roy and Subhas C. Nandy
2012 arXiv   pre-print
., R_n} of n randomly positioned axis parallel rectangles in 2D, the problem of computing the minimum clique cover (MCC) and maximum independent set (MIS) for the intersection graph G( R) of the members  ...  Finally we will provide refined greedy algorithms based on a concept of simplicial rectangle.  ...  Nilson [14] proved that the number of geometric clique in G(R) can be at most τ (2, c)φ(R) log c−1 2 (φ(R) + 1), where τ (2, c) is the Gallai number of the pairwise intersecting c-oriented polygons and  ...

Parameterized Complexity of Stabbing Rectangles and Squares in the Plane [chapter]

Michael Dom, Michael R. Fellows, Frances A. Rosamond
2009 Lecture Notes in Computer Science
k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines.  ...  at most two blocks of 1s per row.  ...  We thank Dániel Marx, who pointed us to the approach for proving that Rectangle Stabbing is in W [1] .  ...

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, Bartosz Walczak
2013 Discrete & Computational Geometry
Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number.  ...  However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane.  ...  Otherwise, if φ uses the same set of colors on these two families, then another color must be used on the diagonal D Q , and thus φ uses at least k colors on the sets intersecting the upper probe U P,Q  ...

On a special class of boxicity 2 graphs [article]

Sujoy Kumar Bhore, Dibyayan Chakraborty, Sandip Das, Sagnik Sen
2016 arXiv   pre-print
A 2SIG is an axes-parallel rectangle intersection graph where the rectangles have unit height (that is, length of the side parallel to Y-axis) and intersects either of the two fixed lines, parallel to  ...  We define and study a class of graphs, called 2-stab interval graphs (2SIG), with boxicity 2 which properly contains the class of interval graphs.  ...  The class of boxicity k graphs is the class of graphs with boxicity at most k.  ...

A note on maximum independent sets in rectangle intersection graphs

Timothy M. Chan
2004 Information Processing Letters
Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem.  ...  similar algorithm running in only O(n log n + n∆ k−1 ) time, where ∆ ≤ n denotes the maximum number of rectangles a point can be in.  ...  The number of iterations is at most n/d, since at least d rectangles are removed per iteration.  ...

Independent Sets of Dynamic Rectangles: Algorithms and Experiments [article]

Sujoy Bhore, Guangping Li, Martin Nöllenburg
2020 arXiv   pre-print
We study the maximal independent set (MIS) and maximum independent set (MAX-IS) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs.  ...  We conclude with an algorithm that maintains a 2-approximate MAX-IS for dynamic sets of uniform height and arbitrary width rectangles with O(ωlog n) update time, where ω is the largest number of maximal  ...  : one query in T (C(M i−1 )), which can return at most 12 points and a constant number of queries in T (C i−1 ) with the slab partition of P x .  ...

Visibility Representations of Boxes in 2.5 Dimensions [article]

Alessio Arleo, Carla Binucci, Emilio Di Giacomo, William S. Evans, Luca Grilli, Giuseppe Liotta, Henk Meijer, Fabrizio Montecchiani, Sue Whitesides, Stephen Wismath
2016 arXiv   pre-print
We prove that: (i) Every complete bipartite graph admits a 2.5D-BR; (ii) The complete graph K_n admits a 2.5D-BR if and only if n ≤ 19; (iii) Every graph with pathwidth at most 7 admits a 2.5D-BR, which  ...  We show that an n-vertex graph that admits a 2.5D-GBR has at most 4n - 6 √(n) edges and this bound is tight.  ...  RVRs can exist only for graphs with thickness at most two and with at most 6n − 20 edges [25] .  ...

Covering and Packing of Rectilinear Subdivision [article]

Satyabrata Jana, Supantha Pandit
2018 arXiv   pre-print
(P1) Stabbing-Subdivision: Stab all bounded faces by selecting a minimum number of points in the plane.  ...  (P3) Dominating-Subdivision: Select a minimum size collection of faces such that any other face has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected faces.  ...  Lemma 2 [13] Let G = (R ∪ B, E) be a bipartite planar graph on red and blue vertex sets R and B, |R| ≥ 2, such that for every subset B ′ ⊆ B of size at most k, where k is a large enough number, |N G  ...

Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity

Timothy M. Chan, Thomas C. Van Dijk, Krzysztof Fleszar, Joachim Spoerhase, Alexander Wolff, Michael Wagner
2018 International Symposium on Algorithms and Computation
A line segment stabs a rectangle if it intersects its left and its right boundary.  ...  Figure 1 An instance of Stabbing (rectangles) with an optimal solution (gray line segments).  ...  We want to count the number of rectangles that are stabbed by at most two segments in S. Consider any i and j satisfying 1 ≤ i ≤ m/2 < j ≤ m.  ...

Witness Rectangle Graphs

Boris Aronov, Muriel Dulieu, Ferran Hurtado
2013 Graphs and Combinatorics
Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.  ...  In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open rectangle with x and y as opposite corners  ...  A WRG has at most two non-trivial connected components. If there are exactly two, each has diameter at most three. If there is one, its diameter is at most six.  ...

2D Fractional Cascading on Axis-aligned Planar Subdivisions [article]

Peyman Afshani, Pingan Cheng
2020 arXiv   pre-print
In the problem, the input is a graph G with constant degree and a set of values for every vertex of G.  ...  We present a number of upper and lower bounds which reveal that in 2D, the problem has a much richer structure.  ...  on trees of height at most h 2 .  ...
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