arXiv subjects

Dibyayan Chakraborty

Publications and source records attributed to Dibyayan Chakraborty.

26 records · Page 2Linked to original sources

arXiv2022

Twin-width VIII: delineation and win-wins

We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures $\mathcal D$ of subclasses of $\mathcal C$, FO model checking is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we show that segment graphs, directed path graphs, and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that $K_{t,t}$-free segment graphs, and axis-parallel $H_t$-free unit segment graphs have bounded twin-width, where $H_t$ is the half-graph or ladder of height $t$. In contrast, axis-parallel $H_4$-free two-lengthed segment graphs have unbounded twin-width. Our new results, combined with the known FPT algorithm for FO model checking on graphs given with $O(1)$-sequences, lead to win-win arguments. For instance, we derive FPT algorithms for $k$-Ladder on visibility graphs of 1.5D terrains, and $k$-Independent Set on visibility graphs of simple polygons.

cs.DS↗

arXiv2021

Finding Geometric Representations of Apex Graphs is NP-Hard

Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{ç}alves, 2009), \textsc{L}-shapes (Gon{ç}alves et al, 2018). For general graphs, however, even deciding whether such representations exist is often $NP$-hard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is $NP$-hard. More precisely, we show that for every positive integer $k$, recognizing every graph class $\mathcal{G}$ which satisfies $\textsc{PURE-2-DIR} \subseteq \mathcal{G} \subseteq \textsc{1-STRING}$ is $NP$-hard, even when the input graphs are apex graphs of girth at least $k$. Here, $PURE-2-DIR$ is the class of intersection graphs of axis-parallel line segments (where intersections are allowed only between horizontal and vertical segments) and \textsc{1-STRING} is the class of intersection graphs of simple curves (where two curves share at most one point) in the plane. This partially answers an open question raised by Kratochv{\'ı}l \& Pergel (2007). Most known $NP$-hardness reductions for these problems are from variants of 3-SAT. We reduce from the \textsc{PLANAR HAMILTONIAN PATH COMPLETION} problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric graphs.

cs.CG↗

arXiv2019

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

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{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 time algorithm to solve the \textsc{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 $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs.

cs.DM↗

arXiv2018

Approximating Minimum Dominating Set on String Graphs

In this paper, we give approximation algorithms for the \textsc{Minimum Dominating Set (MDS)} problem on \emph{string} graphs and its subclasses. A \emph{path} is a simple curve made up of alternating horizontal and vertical line segments. A \emph{$k$-bend path} is a path made up of at most $k + 1$ line segments. An \textsc{L}-path is a $1$-bend path having the shape `\textsc{L}'. A \emph{vertically-stabbed-\textsc{L} graph} is an intersection graph of \textsc{L}-paths intersecting a common vertical line. We give a polynomial time $8$-approximation algorithm for \textsc{MDS} problem on vertically-stabbed-\textsc{L} graphs whose APX-hardness was shown by Bandyapadhyay et al. (\textsc{MFCS}, 2018). To prove the above result, we needed to study the \emph{Stabbing segments with rays} (\textsc{SSR}) problem introduced by Katz et al. (\textsc{Comput. Geom. 2005}). In the \textsc{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 $O((n+m)\log (n+m))$-time $2$-approximation algorithm for the \textsc{SSR} problem where $n$ and $m$ are the number of rays and segments in the input. A \emph{unit $k$-bend path} is a $k$-bend path whose segments are of unit length. A graph is a \emph{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 \textsc{SSR}-problem, we give a polynomial time $O(k^4)$-approximation algorithm for \textsc{MDS} problem on unit $B_k$-VPG graphs for $k\geq 0$.

cs.DM↗

arXiv2018

On bounds on bend number of split and cocomparability graphs

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 known 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. ({\textsc{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 $m, t \in \mathbb{N}$, there exists a cocomparability graph $G_{t,m}$ with $t < bend(G_{t,m}) \leq 4t+29$ 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. ({\textsc{wg 2012}}) who asked whether $B_k$-VPG-chordal $\subsetneq$ $B_{k+1}$-VPG-chordal for all $k \in \mathbb{N}$. We address this by proving that there are infinitely many $m \in \mathbb{N}$ such that $B_m$-VPG-split $\subsetneq$ $B_{m+1}$-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all $t \in \mathbb{N}$, $B_t$-VPG-$Forb(C_{\geq 5})$ $\subsetneq$ $B_{4t+29}$-VPG-$Forb(C_{\geq 5})$, where $Forb(C_{\geq 5})$ denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all $t \in \mathbb{N}$, $PB_t$-VPG-split $\subsetneq PB_{36t+80}$-VPG-split, where $PB_t$-VPG denotes the class of graphs with proper bend number at most $t$.

cs.DM↗

arXiv2018

On the stab number of rectangle intersection graphs

We introduce the notion of \emph{stab number} and \emph{exact stab number} of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. A graph $G$ is said to be a \emph{$k$-stabbable rectangle intersection graph}, or \emph{$k$-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 of the $k$ horizontal lines, then the graph $G$ is said to be a \emph{$k$-exactly stabbable rectangle intersection graph}, or \emph{$k$-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 $k\leq 3$, $k$-SRIG is equivalent to $k$-ESRIG and for any $k\geq 10$, 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 $k\geq 4$.

cs.DM↗

arXiv2016

On a special class of boxicity 2 graphs

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 rule. 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.

cs.DM↗

arXiv2016

On local structures of cubicity 2 graphs

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 NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation.

cs.DM↗