arXiv subjects

Arti Pandey

Publications and source records attributed to Arti Pandey.

17 recordsLinked to original sources

arXiv2025

Vertex-edge domination on subclasses of bipartite graphs

Given a simple undirected graph $G = (V, E)$, the open neighbourhood of a vertex $v \in V$ is defined as $N_G(v) = \{u \in V \mid uv \in E\}$, and the closed neighbourhood as $N_G[v] = N_G(v) \cup \{v\}$. A subset $D \subseteq V$ is called a vertex-edge dominating set if, for every edge $uv \in E$, at least one vertex from $D$ appears in $N_G[u] \cup N_G[v]$; that is, $\vert (N_G[u] \cup N_G[v]) \cap D\vert \geq 1$. Intuitively, a vertex-edge dominating set ensures that every edge, as well as all edges incident to either of its endpoints, is dominated by at least one vertex from the set. The \textsc{Min-VEDS} problem asks for a vertex-edge dominating set of minimum size in a given graph. This problem is known to be NP-complete even for bipartite graphs. In this paper, we strengthen this hardness result by proving that the problem remains NP-complete for two specific subclasses of bipartite graphs: star-convex and comb-convex bipartite graphs. For a graph $G$ on $n$ vertices, it is known that the \textsc{Min-VEDS} problem cannot be approximated within a factor of $(1 - ε)\ln |V|$ for any $ε> 0$, unless $\text{NP} \subseteq \text{DTIME}(|V|^{O(\log \log |V|)})$. We also prove that this inapproximability result holds even for star-convex and comb-convex bipartite graphs. On the positive side, we present a polynomial-time algorithm for computing a minimum vertex-edge dominating set in convex bipartite graphs. A polynomial-time algorithm for this graph class was also proposed by B{ü}y{ü}k{ç}olak et al.~\cite{buyukccolak2025linear}, but we show that their algorithm has certain flaws by providing instances where it fails to produce an optimal solution. We address this issue by presenting a modified algorithm that correctly computes an optimal solution.

math.CO↗

arXiv2025

Edge open packing: further characterizations

Let $G=(V, E)$ be a graph where $V(G)$ and $E(G)$ are the vertex and edge sets, respectively. In a graph $G$, two edges $e_1, e_2\in E(G)$ are said to have \emph{common edge} $e\neq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ in $G$. A subset $D\subseteq E(G)$ is called an \emph{edge open packing set} in $G$ if no two edges in $D$ share a common edge in $G$, and the largest size of such a set in $G$ is known as \emph{edge open packing number}, represented by $ρ_{e}^o(G)$. In the introductory paper (Chelladurai et al. (2022)), necessary and sufficient conditions for $ρ_{e}^o(G)=1, 2$ were provided, and the graphs $G$ with $ρ_{e}^o(G)\in \{m-2, m-1, m\}$ were characterized, where $m$ is the number of edges of $G$. In this paper, we further characterize the graphs $G$. First, we show necessary and sufficient conditions for $ρ_{e}^o(G)=t$, for any integer $t\geq 3$. Finally, we characterize the graphs with $ρ_{e}^o(G)=m-3$.

math.CO↗

arXiv2025

Characterizing optimal monitoring edge-geodetic sets for some structured graph classes

Given a graph $G=(V,E)$, a set $S\subseteq V$ is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in $S$. The minimum size of such a set in $G$ is called the monitoring edge-geodetic number of $G$ and is denoted by $meg(G)$. In this work, we compute the monitoring edge-geodetic number efficiently for the following graph classes: distance-hereditary graphs, $P_4$-sparse graphs, bipartite permutation graphs, and strongly chordal graphs. The algorithms follow from structural characterizations of the optimal monitoring edge-geodetic sets for these graph classes in terms of \emph{mandatory vertices} (those that need to be in every solution). This extends previous results from the literature for cographs, interval graphs and block graphs.

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↗

arXiv2024

Exploring Algorithmic Solutions for the Independent Roman Domination Problem in Graphs

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} 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} (or 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{Independent Roman Domination Number} of $G$, denoted by $i_R(G)$, is defined as min$\{w(f)~\vert~f$ is an IRDF of $G\}$. For a given graph $G$, the problem of computing $i_R(G)$ is defined as the \emph{Minimum Independent Roman Domination problem}. The problem is already known to be NP-hard for bipartite graphs. In this paper, we further study the algorithmic complexity of the problem. In this paper, we propose a polynomial-time algorithm to solve the Minimum Independent Roman Domination problem for distance-hereditary graphs, split graphs, and $P_4$-sparse graphs.

math.CO↗

arXiv2024

Algorithmic Results for Weak Roman Domination Problem in Graphs

Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a \emph{Weak Roman Dominating function} (WRD function) if, for every vertex $u$ with $f(u) = 0$, there exists a neighbour $v$ of $u$ with $f(v) > 0$ and a new function $f': V \rightarrow \{0, 1, 2\}$ defined in the following way: $f'(u) = 1$, $f'(v) = f(v) - 1$, and $f'(w) = f(w)$, for all vertices $w$ in $V\setminus\{u,v\}$; so that no vertices are undefended by $f'$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Weak Roman Domination Number} denoted by $γ_r(G)$, represents $min\{w(f)~\vert~f$ is a WRD function of $G\}$. For a given graph $G$, the problem of finding a WRD function of weight $γ_r(G)$ is defined as the \emph{Minimum Weak Roman domination problem}. The problem is already known to be NP-hard for bipartite and chordal graphs. In this paper, we further study the algorithmic complexity of the problem. We prove the NP-hardness of the problem for star convex bipartite graphs and comb convex bipartite graphs, which are subclasses of bipartite graphs. In addition, we show that for the bounded degree star convex bipartite graphs, the problem is efficiently solvable. We also prove the NP-hardness of the problem for split graphs, a subclass of chordal graphs. On the positive side, we give polynomial-time algorithms to solve the problem for $P_4$-sparse graphs. Further, we have presented some approximation results.

cs.DM↗

arXiv2023

Computation of Grundy dominating sequences in (co-)bipartite graphs

A sequence $S$ of vertices of a graph $G$ is called a dominating sequence of $G$ if $(i)$ each vertex $v$ of $S$ dominates a vertex of $G$ that was not dominated by any of the vertices preceding vertex $v$ in $S$, and $(ii)$ every vertex of $G$ is dominated by at least one vertex of $S$. The Grundy Domination problem is to find a longest dominating sequence for a given graph $G$. It has been known that the decision version of the Grundy Domination problem is NP-complete even when restricted to chordal graphs. In this paper, we prove that the decision version of the Grundy Domination problem is NP-complete for bipartite graphs and co-bipartite graphs. On the positive side, we present a linear-time algorithm that solves the Grundy Domination problem for chain graphs, which form a subclass of bipartite graphs.

math.CO↗

arXiv2023

Complexity of total dominator coloring in graphs

Let $G=(V,E)$ be a graph with no isolated vertices. A vertex $v$ totally dominate a vertex $w$ ($w \ne v$), if $v$ is adjacent to $w$. A set $D \subseteq V$ called a total dominating set of $G$ if every vertex $v\in V$ is totally dominated by some vertex in $D$. The minimum cardinality of a total dominating set is the total domination number of $G$ and is denoted by $γ_t(G)$. A total dominator coloring of graph $G$ is a proper coloring of vertices of $G$, so that each vertex totally dominates some color class. The total dominator chromatic number $χ_{td}(G)$ of $G$ is the least number of colors required for a total dominator coloring of $G$. The Total Dominator Coloring problem is to find a total dominator coloring of $G$ using the minimum number of colors. It is known that the decision version of this problem is NP-complete for general graphs. We show that it remains NP-complete even when restricted to bipartite, planar and split graphs. We further study the Total Dominator Coloring problem for various graph classes, including trees, cographs and chain graphs. First, we characterize the trees having $χ_{td}(T)=γ_t(T)+1$, which completes the characterization of trees achieving all possible values of $χ_{td}(T)$. Also, we show that for a cograph $G$, $χ_{td}(G)$ can be computed in linear-time. Moreover, we show that $2 \le χ_{td}(G) \le 4$ for a chain graph $G$ and give characterization of chain graphs for every possible value of $χ_{td}(G)$ in linear-time.

cs.DM↗

arXiv2023

Cosecure Domination: Hardness Results and Algorithm

For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision version of the problem is known to be NP-complete for bipartite, planar, and split graphs. Also, it is known that the Minimum Cosecure Domination problem is efficiently solvable for proper interval graphs and cographs. In this paper, we work on various important graph classes in an effort to reduce the complexity gap of the Minimum Cosecure Domination problem. We show that the decision version of the problem remains NP-complete for circle graphs, doubly chordal graphs, chordal bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs. On the positive side, we give an efficient algorithm to compute the cosecure domination number of chain graphs, which is an important subclass of bipartite graphs. In addition, we show that the problem is linear-time solvable for bounded tree-width graphs. Further, we prove that the computational complexity of this problem varies from the domination problem.

cs.DM↗

arXiv2021

Algorithms for Maximum Internal Spanning Tree Problem for Some Graph Classes

For a given graph $G$, a maximum internal spanning tree of $G$ is a spanning tree of $G$ with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

cs.DS↗

arXiv2021

Complexity of Paired Domination in AT-free and Planar Graphs

For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired dominating set (PD-set), if $G[D]$, the subgraph induced by $D$ in $G$ has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem remains NP-complete even when $G$ belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time $2$-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NP-complete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput. Sci., $591 (2015): 99-105$ and Algorithmica, $ 82 (2020) :2809-2840$).

cs.DM↗

arXiv2021

A linear-time algorithm for semitotal domination in strongly chordal graphs

In a graph $G=(V,E)$ with no isolated vertex, a dominating set $D \subseteq V$, is called a semitotal dominating set if for every vertex $u \in D$ there is another vertex $v \in D$, such that distance between $u$ and $v$ is at most two in $G$. Given a graph $G=(V,E)$ without isolated vertices, the Minimum Semitotal Domination problem is to find a minimum cardinality semitotal dominating set of $G$. The semitotal domination number, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The decision version of the problem remains NP-complete even when restricted to chordal graphs, chordal bipartite graphs, and planar graphs. Galby et al. in [6] proved that the problem can be solved in polynomial time for bounded MIM-width graphs which includes many well known graph classes, but left the complexity of the problem in strongly chordal graphs unresolved. Henning and Pandey in [20] also asked to resolve the complexity status of the problem in strongly chordal graphs. In this paper, we resolve the complexity of the problem in strongly chordal graphs by designing a linear-time algorithm for the problem.

math.CO↗

arXiv2021

Semipaired Domination in Some Subclasses of Chordal Graphs

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $γ_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $γ_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

cs.DM↗

arXiv2019

Complexity and Algorithms for Semipaired Domination in Graphs

For a graph $G=(V,E)$ with no isolated vertices, a set $D\subseteq V$ is called a semipaired dominating set of G if $(i)$ $D$ is a dominating set of $G$, and $(ii)$ $D$ can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of $G$ is called the semipaired domination number of $G$, and is denoted by $γ_{pr2}(G)$. The \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $γ_{pr2}(G)$. In this paper, we initiate the algorithmic study of the \textsc{Minimum Semipaired Domination} problem. We show that the decision version of the \textsc{Minimum Semipaired Domination} problem is NP-complete for bipartite graphs and split graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs and trees. We also propose a $1+\ln(2Δ+2)$-approximation algorithm for the \textsc{Minimum Semipaired Domination} problem, where $Δ$ denote the maximum degree of the graph and show that the \textsc{Minimum Semipaired Domination} problem cannot be approximated within $(1-ε) \ln|V|$ for any $ε> 0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log\log|V|)})$.

cs.DM↗

arXiv2017

Algorithmic Aspects of Semitotal Domination in Graphs

For a graph $G=(V,E)$, a set $D \subseteq V$ is called a semitotal dominating set of $G$ if $D$ is a dominating set of $G$, and every vertex in $D$ is within distance~$2$ of another vertex of~$D$. The \textsc{Minimum Semitotal Domination} problem is to find a semitotal dominating set of minimum cardinality. Given a graph $G$ and a positive integer $k$, the \textsc{Semitotal Domination Decision} problem is to decide whether $G$ has a semitotal dominating set of cardinality at most $k$. The \textsc{Semitotal Domination Decision} problem is known to be NP-complete for general graphs. In this paper, we show that the \textsc{Semitotal Domination Decision} problem remains NP-complete for planar graphs, split graphs and chordal bipartite graphs. We give a polynomial time algorithm to solve the \textsc{Minimum Semitotal Domination} problem in interval graphs. We show that the \textsc{Minimum Semitotal Domination} problem in a graph with maximum degree~$Δ$ admits an approximation algorithm that achieves the approximation ratio of $2+3\ln(Δ+1)$, showing that the problem is in the class log-APX. We also show that the \textsc{Minimum Semitotal Domination} problem cannot be approximated within $(1 - ε)\ln |V| $ for any $ε> 0$ unless NP $\subseteq$ DTIME $(|V|^{O(\log \log |V|)})$. Finally, we prove that the \textsc{Minimum Semitotal Domination} problem is APX-complete for bipartite graphs with maximum degree $4$.

cs.DM↗

arXiv2016

Some Algorithmic Results on Restrained Domination in Graphs

A set $D\subseteq V$ of a graph $G=(V,E)$ is called a restrained dominating set of $G$ if every vertex not in $D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textsc{Minimum Restrained Domination} problem is to find a restrained dominating set of minimum cardinality. Given a graph $G$, and a positive integer $k$, the \textsc{Restrained Domination Decision} problem is to decide whether $G$ has a restrained dominating set of cardinality a most $k$. The \textsc{Restrained Domination Decision} problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this NP-completeness result by showing that the \textsc{Restrained Domination Decision} problem remains NP-complete for doubly chordal graphs, a subclass of chordal graphs. We also propose a polynomial time algorithm to solve the \textsc{Minimum Restrained Domination} problem in block graphs, a subclass of doubly chordal graphs. The \textsc{Restrained Domination Decision} problem is also known to be NP-complete for split graphs. We propose a polynomial time algorithm to compute a minimum restrained dominating set of threshold graphs, a subclass of split graphs. In addition, we also propose polynomial time algorithms to solve the \textsc{Minimum Restrained Domination} problem in cographs and chain graphs. Finally, we give a new improved upper bound on the restrained domination number, cardinality of a minimum restrained dominating set in terms of number of vertices and minimum degree of graph. We also give a randomized algorithm to find a restrained dominating set whose cardinality satisfy our upper bound with a positive probability.

cs.DM↗

arXiv2015

Algorithmic aspects of disjunctive domination in graphs

For a graph $G=(V,E)$, a set $D\subseteq V$ is called a \emph{disjunctive dominating set} of $G$ if for every vertex $v\in V\setminus D$, $v$ is either adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it. The cardinality of a minimum disjunctive dominating set of $G$ is called the \emph{disjunctive domination number} of graph $G$, and is denoted by $γ_{2}^{d}(G)$. The \textsc{Minimum Disjunctive Domination Problem} (MDDP) is to find a disjunctive dominating set of cardinality $γ_{2}^{d}(G)$. Given a positive integer $k$ and a graph $G$, the \textsc{Disjunctive Domination Decision Problem} (DDDP) is to decide whether $G$ has a disjunctive dominating set of cardinality at most $k$. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a $(\ln(Δ^{2}+Δ+2)+1)$-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within $(1-ε) \ln(|V|)$ for any $ε>0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log \log |V|)})$. Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree $3$.

cs.DM↗