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On the stab number of rectangle intersection graphs
[article]

2018
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arXiv
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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

arXiv:1804.06571v1
fatcat:m23w5dsxxram7ibms7bipswy2i
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... 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.##
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On local structures of cubicity 2 graphs
[article]

2016
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arXiv
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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

arXiv:1603.09570v1
fatcat:pohuogwanzeqfp4ibparjjhqmi
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... We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation.##
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On a special class of boxicity 2 graphs
[article]

2016
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arXiv
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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

arXiv:1603.09561v1
fatcat:phzx2lo6dbardfzll7nvwny2vy
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... . 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.##
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Approximating Minimum Dominating Set on String Graphs
[article]

2018
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arXiv
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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

arXiv:1809.09990v2
fatcat:viiuk4eguzandknoijta7sko7e
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... 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.##
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Hardness and approximation for the geodetic set problem in some graph classes
[article]

2019
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arXiv
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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

arXiv:1909.08795v1
fatcat:fpe4dfo6l5f4xltxjwescdepk4
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... 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.##
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Algorithms and complexity for geodetic sets on planar and chordal graphs
[article]

2020
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arXiv
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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

arXiv:2006.16511v1
fatcat:ax56pso6ynb4bmc3jb2yf6rt6i
*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. ...##
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On bounds on bend number of split and cocomparability graphs
[article]

2018
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arXiv
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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

arXiv:1804.06584v2
fatcat:qpqstg5mgbfcjc4axr2tq7bsiy
## 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.##
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Algorithms and Complexity for Geodetic Sets on Planar and Chordal Graphs

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

doi:10.4230/lipics.isaac.2020.7
fatcat:e6s4ypgscvd5vgfdcjbeltsvbm
*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. ...##
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A novel method for segmenting and straightening of text lines in handwritten Telugu documents based on smearing and regression approach

2018
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International Journal of Engineering & Technology
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In [8]

doi:10.14419/ijet.v7i3.13286
fatcat:gxvrm3yq7zdvlcnqw3kuoqrsq4
*Dibyayan**Chakraborty*proposed method for detecting base line from multi-lingual multi-turn handwritten document images. ...##
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Pseudoline arrangement graphs: degree sequences and eccentricities
[article]

2021
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arXiv
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pre-print

The authors also thank

arXiv:2103.02283v1
fatcat:fmpde3vwsbfr3bzxfbwg252mfq
*Dibyayan**Chakraborty*for suggesting to pursue the questions on eccentricity. ...