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### On the stab number of rectangle intersection graphs [article]

Dibyayan Chakraborty, Mathew C. Francis
2018 arXiv   pre-print
A lower bound on the stab number of rectangle intersection graphs in terms of its pathwidth and clique number is shown.  ...  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 Section 4, we show a lower bound on the stab number of rectangle intersection graphs in terms of the clique number and the pathwidth, and then study upper bounds on the stab number of rectangle intersection  ...

### 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 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.  ...  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  ...  Proximity graph  is a graph where the edges between the vertices of the graphs depend on the neighbourlines of the vertices.  ...

### 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.  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s)  ...

### 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.  ...  Thus, the minimum clique cover of the graph G(R) is same as the minimum number of points required to stab all the rectangles inR.  ...

### 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.  ...  Now consider axes-parallel rectangles with unit height (length of the side parallel to Y -axis) that intersects one of the stab lines.  ...

### 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.  ...  Recall that a rectangle is stabbed by a line if their intersection is nonempty.  ...

### 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  ...  Section 3 gives an overview of the method we use to solve Cochromatic Number on perfect graphs and Disjoint Rectangle Stabbing.  ...

### 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.  ...  For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W-hardness, while the parameterized complexity of the special case where the input consists  ...  We thank Dániel Marx, who pointed us to the approach for proving that Rectangle Stabbing is in W  .  ...

### 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.  ...  , or equivalently, find a maximum independent set in the intersection graph of the rectangles.  ...

### 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  ...  ., that every rectangle in R is stabbed by exactly one line in H. For a set of rectangles R, we denote the subset stabbed by a line h j as R(h j ) ⊆ R.  ...

### Extension complexity of stable set polytopes of bipartite graphs [article]

Manuel Aprile, Yuri Faenza, Samuel Fiorini, Tony Huynh, Marco Macchia
2017 arXiv   pre-print
The extension complexity xc(P) of a polytope P is the minimum number of facets of a polytope that affinely projects to P.  ...  Let G be a bipartite graph with n vertices, m edges, and no isolated vertices. Let STAB(G) be the convex hull of the stable sets of G. It is easy to see that n ≤xc (STAB(G)) ≤ n+m.  ...  We thank Monique Laurent and Ronald de Wolf for bringing the topic of this paper to our attention.  ...

### All-Pairs Shortest Paths in Geometric Intersection Graphs

Timothy M. Chan, Dimitrios Skrepetos
2018 Journal of Computational Geometry
Specifically we reduce the problem to the design of static data structures for offline intersection detection.  ...  Thus, we obtain an \$\OO{n \log n}\$-time SSSP algorithm in unweighted intersection graphs of \$n\$ axis-aligned line segments.  ...  To derive our result for intersection graphs of axis-aligned boxes, we describe in Section 5 a new O n √ log n -time algorithm for oine rectangle stabbing in two dimensions: preprocess n axis-aligned rectangles  ...

### 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.  ...  We study a class of geometric covering and packing problems for bounded regions on the plane.  ...  In  , Gaur et al. studied the rectangle stabbing problem. Here  given a set of rectangles, the objective is to stab all rectangles with a minimum number of axis-parallel lines.  ...

### 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
Finally, we prove that deciding whether a given graph G admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR Γ is the set of bottom faces of the boxes in Γ.  ...  We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane z=0 and edges are unobstructed lines of sight parallel to  ...  We now prove that if h and v both stab ten boxes, there must be one box that is stabbed by both h and v , which implies that the number of boxes in Γ is at most 19.  ...

### Extended formulations from communication protocols in output-efficient time [article]

Manuel Aprile, Yuri Faenza
2019 arXiv   pre-print
Although constructive, those tools are not output-efficient, since the time needed to produce the extended formulation also depends on the number of rows of the slack matrix (hence, on the exact description  ...  ~Yannakakis' extended formulation for the stable set polytope of perfect graphs, for which, to the best of our knowledge, an efficient construction was previously not known.  ...  We thank Mihalis Yannakakis for inspiring discussions and Samuel Fiorini for useful comments on  , where many of the results here presented appeared.  ...
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