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On the stab number of rectangle intersection graphs [article]

Dibyayan Chakraborty, Mathew C. Francis
2018 arXiv   pre-print
We introduce the notion of stab number and exact stab number of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. A graph G is said to be a k-stabbable rectangle intersection graph, or k-SRIG for short, if it has a rectangle intersection representation in which k horizontal lines can be chosen such that each rectangle is intersected by at least one of them. If there exists such a representation with the additional property that each rectangle intersects exactly one
more » ... of the k horizontal lines, then the graph G is said to be a k-exactly stabbable rectangle intersection graph, or k-ESRIG for short. The stab number of a graph G, denoted by stab(G), is the minimum integer k such that G is a k-SRIG. Similarly, the exact stab number of a graph G, denoted by estab(G), is the minimum integer k such that G is a k-ESRIG. In this work, we study the stab number and exact stab number of some subclasses of rectangle intersection graphs. A lower bound on the stab number of rectangle intersection graphs in terms of its pathwidth and clique number is shown. Tight upper bounds on the exact stab number of split graphs with boxicity at most 2 and block graphs are also given. We show that for k≤ 3, k-SRIG is equivalent to k-ESRIG and for any k≥ 10, there is a tree which is a k-SRIG but not a k-ESRIG. We also develop a forbidden structure characterization for block graphs that are 2-ESRIG and trees that are 3-ESRIG, which lead to polynomial-time recognition algorithms for these two classes of graphs. These forbidden structures are natural generalizations of asteroidal triples. Finally, we construct examples to show that these forbidden structures are not sufficient to characterize block graphs that are 3-SRIG or trees that are k-SRIG for any k≥ 4.
arXiv:1804.06571v1 fatcat:m23w5dsxxram7ibms7bipswy2i

On local structures of cubicity 2 graphs [article]

Sujoy Kumar Bhore, Dibyayan Chakraborty, Sandip Das, Sagnik Sen
2016 arXiv   pre-print
A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the X-axis, distance 1 + ϵ (0 < ϵ < 1) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be
more » ... We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation.
arXiv:1603.09570v1 fatcat:pohuogwanzeqfp4ibparjjhqmi

On a special class of boxicity 2 graphs [article]

Sujoy Kumar Bhore, Dibyayan Chakraborty, Sandip Das, Sagnik Sen
2016 arXiv   pre-print
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. 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 the X-axis, distance 1+ϵ (0 < ϵ < 1) apart. Intuitively, 2SIG is a graph obtained by putting some edges between two interval graphs in a particular
more » ... . It turns out that for these kind of graphs, the chromatic number of any of its induced subgraphs is bounded by twice of its (induced subgraph) clique number. This shows that the graph, even though not perfect, is not very far from it. Then we prove similar results for some subclasses of 2SIG and provide efficient algorithm for finding their clique number. We provide a matrix characterization for a subclass of 2SIG graph.
arXiv:1603.09561v1 fatcat:phzx2lo6dbardfzll7nvwny2vy

Approximating Minimum Dominating Set on String Graphs [article]

Dibyayan Chakraborty, Sandip Das, Joydeep Mukherjee
2018 arXiv   pre-print
In this paper, we give approximation algorithms for the Minimum Dominating Set (MDS) problem on string graphs and its subclasses. A path is a simple curve made up of alternating horizontal and vertical line segments. A k-bend path is a path made up of at most k + 1 line segments. An L-path is a 1-bend path having the shape 'L'. A vertically-stabbed-L graph is an intersection graph of L-paths intersecting a common vertical line. We give a polynomial time 8-approximation algorithm for MDS problem
more » ... on vertically-stabbed-L graphs whose APX-hardness was shown by Bandyapadhyay et al. (MFCS, 2018). To prove the above result, we needed to study the Stabbing segments with rays (SSR) problem introduced by Katz et al. (Comput. Geom. 2005). In the SSR problem, the input is a set of (disjoint) leftward-directed rays, and a set of (disjoint) vertical segments. The objective is to select a minimum number of rays that intersect all vertical segments. We give a O((n+m) (n+m))-time 2-approximation algorithm for the SSR problem where n and m are the number of rays and segments in the input. A unit k-bend path is a k-bend path whose segments are of unit length. A graph is a unit B_k-VPG graph if it is an intersection graph of unit k-bend paths. Any string graph is a unit-B_k-VPG graph for some finite k. Using our result on SSR-problem, we give a polynomial time O(k^4)-approximation algorithm for MDS problem on unit B_k-VPG graphs for k≥ 0.
arXiv:1809.09990v2 fatcat:viiuk4eguzandknoijta7sko7e

Hardness and approximation for the geodetic set problem in some graph classes [article]

Dibyayan Chakraborty, Florent Foucaud, Harmender Gahlawat, Subir Kumar Ghosh, Bodhayan Roy
2019 arXiv   pre-print
In this paper, we study the computational complexity of finding the geodetic number of graphs. A set of vertices S of a graph G is a geodetic set if any vertex of G lies in some shortest path between some pair of vertices from S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the MGS problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless P=NP, there is no polynomial
more » ... ime algorithm to solve the MGS problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter 2. On the positive side, we give an O(√(n)log n)-approximation algorithm for the MGS problem on general graphs of order n. We also give a 3-approximation algorithm for the MGS problem on the family of solid grid graphs which is a subclass of planar graphs.
arXiv:1909.08795v1 fatcat:fpe4dfo6l5f4xltxjwescdepk4

Algorithms and complexity for geodetic sets on planar and chordal graphs [article]

Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, Bodhayan Roy
2020 arXiv   pre-print
We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al.  ...  Chakraborty et al. [6] proved the following lemma. Lemma 5 ([6]). Any geodetic set of G contains at least one vertex from each corner path.  ... 
arXiv:2006.16511v1 fatcat:ax56pso6ynb4bmc3jb2yf6rt6i

On bounds on bend number of split and cocomparability graphs [article]

Dibyayan Chakraborty, Sandip Das, Joydeep Mukherjee, Uma kant Sahoo
2018 arXiv   pre-print
A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A k-bend path is a path made up of at most k + 1 line segments. A B_k-VPG representation of a graph is a collection of k-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a B_k-VPG representation are called B_k-VPG graphs. It is
more » ... nown that the poset dimension dim(G) of a cocomparability graph G is greater than or equal to its bend number bend(G). Cohen et al. (order 2015) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each m, t ∈N, there exists a cocomparability graph G_t,m with t < bend(G_t,m) ≤ 4t+29 and dim(G_t,m)-bend(G_t,m)>m. Techniques used to prove the above result, allows us to partially address the open question posed by Chaplick et al. (wg 2012) who asked whether B_k-VPG-chordal ⊊ B_k+1-VPG-chordal for all k ∈N. We address this by proving that there are infinitely many m ∈N such that B_m-VPG-split ⊊ B_m+1-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all t ∈N, B_t-VPG-Forb(C_≥ 5) ⊊ B_4t+29-VPG-Forb(C_≥ 5), where Forb(C_≥ 5) denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all t ∈N, PB_t-VPG-split ⊊ PB_36t+80-VPG-split, where PB_t-VPG denotes the class of graphs with proper bend number at most t.
arXiv:1804.06584v2 fatcat:qpqstg5mgbfcjc4axr2tq7bsiy

Algorithms and Complexity for Geodetic Sets on Planar and Chordal Graphs

Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, Bodhayan Roy, Yixin Cao, Siu-Wing Cheng, Minming Li
2020
We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al.  ...  Chakraborty et al. [6] proved: Lemma 5 ([6] ). Any geodetic set of G contains at least one vertex from each corner path. Any geodetic set of G contains all vertices of degree 0 or 1.  ... 
doi:10.4230/lipics.isaac.2020.7 fatcat:e6s4ypgscvd5vgfdcjbeltsvbm

A novel method for segmenting and straightening of text lines in handwritten Telugu documents based on smearing and regression approach

Mslb. Subrahmanyam, V Vijaya Kumar, B Eswara Reddy
2018 International Journal of Engineering & Technology  
In [8] Dibyayan Chakraborty proposed method for detecting base line from multi-lingual multi-turn handwritten document images.  ... 
doi:10.14419/ijet.v7i3.13286 fatcat:gxvrm3yq7zdvlcnqw3kuoqrsq4

Pseudoline arrangement graphs: degree sequences and eccentricities [article]

Sandip Das, Siddani Bhaskara Rao, Uma kant Sahoo
2021 arXiv   pre-print
The authors also thank Dibyayan Chakraborty for suggesting to pursue the questions on eccentricity.  ... 
arXiv:2103.02283v1 fatcat:fmpde3vwsbfr3bzxfbwg252mfq