arXiv subjects

Gautam K. Das

Publications and source records attributed to Gautam K. Das.

16 recordsLinked to original sources

arXiv2026

Maximum Edge Open Packing in Permutation, Interval, and Well-Partitioned Chordal Graphs

Edge open packing is a relaxation of induced matching in which the selected edges may induce disjoint stars. We study the \textsc{Maximum Edge Open Packing} problem on permutation graphs, interval graphs, and well-partitioned chordal graphs. For the first two classes, we introduce an oriented star-conflict graph whose vertices are ordered edges. We prove that its compatibility graph admits a natural transitive orientation: a product-order orientation for permutation graphs and a left-to-right orientation for interval graphs. In each case, a maximum edge open packing is obtained from a maximum clique, equivalently a longest directed path, in the compatibility graph. Given the corresponding representation, both algorithms run in $O(n^2+m^2)\leq O(n^4)$ time, where $n=|V(G)|$ and $m=|E(G)|$. For well-partitioned chordal graphs, we give a dynamic program over a partition tree. Its states use the fact that the endpoint set of an edge open packing meets each clique bag in at most two vertices. Given a partition-tree representation, the edge open packing number is computed in $O(n^4)$ time, and an optimal packing can be reconstructed within the same time bound.

cs.DM↗

arXiv2026

Roman Domination on Circular-Convex, Triad-Convex Bipartite Graphs and $P_4$-Tidy Graphs

The Roman Domination Problem (RDP) on a graph $G=(V,E)$ asks for a labeling function $f:V\rightarrow\{0,1,2\}$ such that every vertex assigned value $0$ is adjacent to a vertex assigned value $2$. The objective is to minimize the total weight $\sum_{v\in V} f(v)$; this minimum value is the Roman domination number of $G$, denoted by $γ_R(G)$. In this paper, we study RDP on graph classes motivated by convexity and induced-$P_4$ structure. First, we consider circular-convex bipartite graphs, a natural superclass of convex bipartite graphs, where RDP is already known to be polynomial-time solvable. Assuming that a circular-convex representation is given, we compute $γ_R(G)$ in $O(n^6)$ time by cutting the circular order, separating interval and wrap-around vertices, and branching over at most two wrap-around vertices assigned value $2$. Second, we study triad-convex bipartite graphs, a restricted subclass of tree-convex bipartite graphs whose convexity tree is a subdivision of $K_{1,3}$. Although RDP is hard on broader tree-convex subclasses such as star-convex and comb-convex bipartite graphs, we show that $γ_R(G)$ can be computed in $O(n^7)$ time on triad-convex bipartite graphs. Finally, we study $P_4$-tidy graphs, which properly extend cographs. Using the Giakoumakis et al. structural decomposition of $P_4$-tidy graphs, we give a direct, exact algorithm that computes $γ_R(G)$ in $O(n+m)$ time. These results extend the algorithmic boundary of Roman domination on convexity-based bipartite graphs and $P_4$-structured graph classes.

cs.DM↗

arXiv2026

Roman-Type Domination on Convex and Chordal Bipartite Graphs: Algorithms and Hardness

Roman domination and its variants form an important family of domination-type graph parameters motivated by protection, fault tolerance, and resource allocation. A Roman dominating function of a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has a neighbour $u$ with $f(u)=2$. The weight of $f$ is $w(f)=\sum_{v\in V(G)}f(v)$, and the minimum weight of a Roman dominating function of $G$ is the Roman domination number, denoted by $γ_R(G)$. In this paper, we study four variants of Roman domination on two natural subclasses of bipartite graphs, namely convex bipartite graphs and chordal bipartite graphs. On the positive side, we develop a unified left-to-right dynamic programming framework for Roman-$\{2\}$ domination, double Roman domination, perfect Roman domination, and unique response Roman domination on convex bipartite graphs. The algorithms exploit the interval structure of one bipartition class and represent all unfinished requirements using a constant number of boundary indices. Consequently, each of the four parameters can be computed in $O(n^6)$ time, where $n=|V(G)|$. On the negative side, we prove that Roman-$\{2\}$ domination, perfect Roman domination, and unique response Roman domination remain NP-complete on chordal bipartite graphs. These results establish a clear algorithmic separation between convex bipartite graphs, where the interval ordering yields polynomial-time solvability, and the broader class of chordal bipartite graphs, where several Roman-type domination problems remain computationally intractable.

cs.DM↗

arXiv2026

Roman Domination in Convex Bipartite Graphs

In the Roman domination problem, an undirected simple graph $G(V,E)$ is given. The objective of Roman domination problem is to find a function $f:V\rightarrow {\{0,1,2\}}$ such that for any vertex $v\in V$ with $f(v)=0$ must be adjacent to at least one vertex $u\in V$ with $f(u)=2$ and $\sum_{u\in V} f(u)$, called Roman domination number, is minimized. It is already proven that the Roman domination problem (RDP) is NP-complete for general graphs and it remains NP-complete for bipartite graphs. In this paper, we propose a dynamic programming based polynomial time algorithm for RDP in convex bipartite graph.

math.CO↗

arXiv2026

Roman domination on subclasses of bipartite graphs

The Roman Domination Problem (RDP) on a simple, finite, undirected graph $G=(V,E)$ asks for a labeling function $f:V\rightarrow\{0,1,2\}$ such that every vertex assigned value $0$ is adjacent to at least one vertex assigned value $2$. The objective is to minimize the total weight $\sum_{v\in V} f(v)$, and this minimum value is called the Roman domination number of $G$, denoted by $γ_R(G)$. Since the RDP is NP-complete for bipartite graphs, a natural direction is to study its complexity on restricted subclasses of bipartite graphs. The problem remains NP-complete even under strong structural restrictions, such as star-convex and comb-convex bipartite graphs. Therefore, identifying the borderline subclasses where the problem changes from NP-complete to polynomial-time solvable remains an important challenge. In this paper, we investigate the RDP on convex bipartite graphs and on their superclass, chordal bipartite graphs. First, we present a dynamic programming algorithm for convex bipartite graphs. The algorithm uses the interval ordering of one bipartition class and keeps a compact boundary state, which is sufficient to control the domination requirements of both processed and future vertices. This gives an $O(n^3)$-time algorithm for computing $γ_R(G)$ on an $n$ vertex convex bipartite graph. In contrast, we prove that the decision version of the RDP is NP-complete on chordal bipartite graphs by a polynomial reduction from \textsc{Dominating Set} on chordal bipartite graphs. Thus, our results show a clear separation between the tractability of convex bipartite graphs and the hardness of the larger chordal bipartite class.

math.CO↗

arXiv2026

Algorithms for the Maximum Edge Open Packing Problem

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device can participate in at most one communication at a time and simultaneous links must avoid interference we consider a generalization of induced matching known as \emph{edge open packing}. Two edges of a graph are said to conflict if a third edge connects one endpoint of each; an \emph{edge open packing set} is a set of edges containing no such conflicting pair. The largest cardinality of such a set is the \emph{edge open packing number} of a graph. In this work, we study the computational complexity of the Maximum Edge Open Packing Problem. We give a polynomial-time algorithm for the problem in \emph{distance-hereditary graphs}, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on \emph{biconvex bipartite graphs}, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for \emph{chordal graphs}, parameterized by the clique number $ω$, running in $O(2^ω\cdot\mathrm{poly}(n))$ time.

math.CO↗

arXiv2024

(Independent) Roman Domination Parameterized by Distance to Cluster

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} (RDF) if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be an \emph{Independent Roman Dominating function} (IRDF), if $V_1\cup V_2$ forms an independent set, where $V_i=\{v\in V~\vert~f(v)=i\}$, for $i\in \{0,1,2\}$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Roman Domination Number} (resp. \emph{Independent Roman Domination Number}) of $G$, denoted by $γ_R(G)$ (resp. $i_R(G)$), is defined as min$\{w(f)~\vert~f$ is an RDF (resp. IRDF) of $G\}$. For a given graph $G$, the problem of computing $γ_R(G)$ (resp. $i_R(G)$) is defined as the \emph{Roman Domination problem} (resp. \emph{Independent Roman Domination problem}). In this paper, we examine structural parameterizations of the (Independent) Roman Domination problem. We propose fixed-parameter tractable (FPT) algorithms for the (Independent) Roman Domination problem in graphs that are $k$ vertices away from a cluster graph. These graphs have a set of $k$ vertices whose removal results in a cluster graph. We refer to $k$ as the distance to the cluster graph. Specifically, we prove the following results when parameterized by the deletion distance $k$ to cluster graphs: we can find the Roman Domination Number (and Independent Roman Domination Number) in time $4^kn^{O(1)}$. In terms of lower bounds, we show that the Roman Domination number can not be computed in time $2^{εk}n^{O(1)}$, for any $0<ε<1$ unless a well-known conjecture, SETH fails. In addition, we also show that the Roman Domination problem parameterized by distance to cluster, does not admit a polynomial kernel unless NP $\subseteq$ coNP$/$poly.

cs.CC↗

arXiv2021

Approximation Algorithms For The Dispersion Problems in a Metric Space

In this article, we consider the $c$-dispersion problem in a metric space $(X,d)$. Let $P=\{p_{1}, p_{2}, \ldots, p_{n}\}$ be a set of $n$ points in a metric space $(X,d)$. For each point $p \in P$ and $S \subseteq P$, we define $cost_{c}(p,S)$ as the sum of distances from $p$ to the nearest $c $ points in $S \setminus \{p\}$, where $c\geq 1$ is a fixed integer. We define $cost_{c}(S)=\min_{p \in S}\{cost_{c}(p,S)\}$ for $S \subseteq P$. In the $c$-dispersion problem, a set $P$ of $n$ points in a metric space $(X,d)$ and a positive integer $k \in [c+1,n]$ are given. The objective is to find a subset $S\subseteq P$ of size $k$ such that $cost_{c}(S)$ is maximized. We propose a simple polynomial time greedy algorithm that produces a $2c$-factor approximation result for the $c$-dispersion problem in a metric space. The best known result for the $c$-dispersion problem in the Euclidean metric space $(X,d)$ is $2c^2$, where $P \subseteq \mathbb{R}^2$ and the distance function is Euclidean distance [ Amano, K. and Nakano, S. I., Away from Rivals, CCCG, pp.68-71, 2018 ]. We also prove that the $c$-dispersion problem in a metric space is $W[1]$-hard.

cs.CG↗

arXiv2021

Approximation Algorithms For The Euclidean Dispersion Problems

In this article, we consider the Euclidean dispersion problems. Let $P=\{p_{1}, p_{2}, \ldots, p_{n}\}$ be a set of $n$ points in $\mathbb{R}^2$. For each point $p \in P$ and $S \subseteq P$, we define $cost_γ(p,S)$ as the sum of Euclidean distance from $p$ to the nearest $γ$ point in $S \setminus \{p\}$. We define $cost_γ(S)=\min_{p \in S}\{cost_γ(p,S)\}$ for $S \subseteq P$. In the $γ$-dispersion problem, a set $P$ of $n$ points in $\mathbb{R}^2$ and a positive integer $k \in [γ+1,n]$ are given. The objective is to find a subset $S\subseteq P$ of size $k$ such that $cost_γ(S)$ is maximized. We consider both $2$-dispersion and $1$-dispersion problem in $\mathbb{R}^2$. Along with these, we also consider $2$-dispersion problem when points are placed on a line. In this paper, we propose a simple polynomial time $(2\sqrt 3 + ε)$-factor approximation algorithm for the $2$-dispersion problem, for any $ε> 0$, which is an improvement over the best known approximation factor $4\sqrt3$ [Amano, K. and Nakano, S. I., An approximation algorithm for the $2$-dispersion problem, IEICE Transactions on Information and Systems, Vol. 103(3), pp. 506-508, 2020]. Next, we develop a common framework for designing an approximation algorithm for the Euclidean dispersion problem. With this common framework, we improve the approximation factor to $2\sqrt 3$ for the $2$-dispersion problem in $\mathbb{R}^2$. Using the same framework, we propose a polynomial time algorithm, which returns an optimal solution for the $2$-dispersion problem when points are placed on a line. Moreover, to show the effectiveness of the framework, we also propose a $2$-factor approximation algorithm for the $1$-dispersion problem in $\mathbb{R}^2$.

cs.CG↗

arXiv2021

On $d$-distance $m$-tuple ($\ell, r$)-domination in graphs

In this article, we study the $d$-distance $m$-tuple ($\ell, r$)-domination problem. Given a simple undirected graph $G=(V, E)$, and positive integers $d, m, \ell$ and $r$, a subset $V' \subseteq V$ is said to be a $d$-distance $m$-tuple ($\ell, r$)-dominating set if it satisfies the following conditions: (i) each vertex $v \in V$ is $d$-distance dominated by at least $m$ vertices in $V'$, and (ii) each $r$ size subset $U$ of $V$ is $d$-distance dominated by at least $\ell$ vertices in $V'$. Here, a vertex $v$ is $d$-distance dominated by another vertex $u$ means the shortest path distance between $u$ and $v$ is at most $d$ in $G$. A set $U$ is $d$-distance dominated by a set of $\ell$ vertices means size of the union of the $d$-distance neighborhood of all vertices of $U$ in $V'$ is at least $\ell$. The objective of the $d$-distance $m$-tuple ($\ell, r$)-domination problem is to find a minimum size subset $V' \subseteq V$ satisfying the above two conditions. We prove that the problem of deciding whether a graph $G$ has (i) a 1-distance $m$-tuple ($\ell, r$)-dominating set for each fixed value of $m, \ell$, and $r$, and (ii) a $d$-distance $m$-tuple ($\ell, 2$)-dominating set for each fixed value of $d (> 1), m$, and $\ell$ of cardinality at most $k$ (here $k$ is a positive integer) are NP-complete. We also prove that for any $\varepsilon>0$, the 1-distance $m$-tuple $(\ell, r)$-domination problem and the $d$-distance $m$-tuple $(\ell,2)$-domination problem cannot be approximated within a factor of $(\frac{1}{2}- \varepsilon)\ln |V|$ and $(\frac{1}{4}- \varepsilon)\ln |V|$, respectively, unless $P = NP$.

cs.CC↗

arXiv2020

Total Domination in Unit Disk Graphs

Let $G=(V,E)$ be an undirected graph. We call $D_t \subseteq V$ as a total dominating set (TDS) of $G$ if each vertex $v \in V$ has a dominator in $D$ other than itself. Here we consider the TDS problem in unit disk graphs, where the objective is to find a minimum cardinality total dominating set for an input graph. We prove that the TDS problem is NP-hard in unit disk graphs. Next, we propose an 8-factor approximation algorithm for the problem. The running time of the proposed approximation algorithm is $O(n \log k)$, where $n$ is the number of vertices of the input graph and $k$ is output size. We also show that TDS problem admits a PTAS in unit disk graphs.

cs.DS↗

arXiv2020

The Generalized Independent and Dominating Set Problems on Unit Disk Graphs

In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.

cs.DS↗

arXiv2020

Liar's Domination in Unit Disk Graphs

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time $\frac{11}{2}$-factor approximation algorithm \cite{bhore} for the MLDS problem is erroneous and propose a simple $O(n + m)$ time 7.31-factor approximation algorithm, where $n$ and $m$ are the number of vertices and edges in the input unit disk graph, respectively. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.

cs.CC↗

arXiv2014

Minimum Dominating Set for a Point Set in $\IR^2$

In this article, we consider the problem of computing minimum dominating set for a given set $S$ of $n$ points in $\IR^2$. Here the objective is to find a minimum cardinality subset $S'$ of $S$ such that the union of the unit radius disks centered at the points in $S'$ covers all the points in $S$. We first propose a simple 4-factor and 3-factor approximation algorithms in $O(n^6 \log n)$ and $O(n^{11} \log n)$ time respectively improving time complexities by a factor of $O(n^2)$ and $O(n^4)$ respectively over the best known result available in the literature [M. De, G.K. Das, P. Carmi and S.C. Nandy, {\it Approximation algorithms for a variant of discrete piercing set problem for unit disk}, Int. J. of Comp. Geom. and Appl., to appear]. Finally, we propose a very important shifting lemma, which is of independent interest and using this lemma we propose a $\frac{5}{2}$-factor approximation algorithm and a PTAS for the minimum dominating set problem.

cs.DS↗

arXiv2012

Unit Disk Cover Problem

Given a set ${\cal D}$ of unit disks in the Euclidean plane, we consider (i) the {\it discrete unit disk cover} (DUDC) problem and (ii) the {\it rectangular region cover} (RRC) problem. In the DUDC problem, for a given set ${\cal P}$ of points the objective is to select minimum cardinality subset ${\cal D}^* \subseteq {\cal D}$ such that each point in ${\cal P}$ is covered by at least one disk in ${\cal D}^*$. On the other hand, in the RRC problem the objective is to select minimum cardinality subset ${\cal D}^{**} \subseteq {\cal D}$ such that each point of a given rectangular region ${\cal R}$ is covered by a disk in ${\cal D}^{**}$. For the DUDC problem, we propose an $(9+ε)$-factor ($0 < ε\leq 6$) approximation algorithm. The previous best known approximation factor was 15 \cite{FL12}. For the RRC problem, we propose (i) an $(9 + ε)$-factor ($0 < ε\leq 6$) approximation algorithm, (ii) an 2.25-factor approximation algorithm in reduce radius setup, improving previous 4-factor approximation result in the same setup \cite{FKKLS07}. The solution of DUDC problem is based on a PTAS for the subproblem LSDUDC, where all the points in ${\cal P}$ are on one side of a line and covered by the disks centered on the other side of that line.

cs.CG↗

arXiv2012

Improved Interference in Wireless Sensor Networks

Given a set ${\cal V}$ of $n$ sensor node distributed on a 2-dimensional plane and a source node $s \in {\cal V}$, the {\it interference problem} deals with assigning transmission range to each $v \in {\cal V}$ such that the members in ${\cal V}$ maintain connectivity predicate ${\cal P}$, and the maximum/total interference is minimum. We propose algorithm for both {\it minimizing maximum interference} and {\it minimizing total interference} of the networks. For minimizing maximum interference we present optimum solution with running time $O(({\cal P}_n + n^2) \log n)$ for connectivity predicate ${\cal P}$ like strong connectivity, broadcast ($s$ is the source), $k$-edge(vertex) connectivity, spanner, where $O({\cal P}_n)$ is the time complexity for checking the connectivity predicate ${\cal P}$. The running time of the previous best known solution was $O({\cal P}_n \times n^2)$ [Bil$\grave{o}$ and Proietti, 2008]. For the minimizing total interference we propose optimum algorithm for the connectivity predicate broadcast. The running time of the propose algorithm is O(n). For the same problem, the previous best known result was $2(1 + \ln (n-1))$-factor approximation algorithm [Bil$\grave{o}$ and Proietti, 2008]. We also propose a heuristic for minimizing total interference in the case of strongly connected predicate and compare our result with the best result available in the literature. Experimental results demonstrate that our heuristic outperform existing result.

cs.NI↗