arXiv subjects

Sweta Das

Publications and source records attributed to Sweta Das.

6 recordsLinked to original sources

arXiv2026

Exact-Distance Domination in Grid Graphs

Let $G_n$ be the $n\times n$ square grid, and let $k\geq 2$. A set $D\subseteq V(G_n)$ is an \emph{exact-distance $k$-dominating set} if every vertex $v\in V(G_n)\setminus D$ has a vertex $u\in D$ with $d(u,v)=k$. We write $D_{\mathrm{opt}}^{(k)}(G_n)$ for the minimum cardinality of such a set. For every fixed $k$, consider the limit $ \delta_k= \lim_{n\to\infty} \frac{D_{\mathrm{opt}}^{(k)}(G_n)}{n^2}. $ We prove that, for every fixed $k\geq 3$, $ \frac{1}{4k} \leq \delta_k \leq \frac{k-1}{3k^2-k-1}. $ For $k=2$, the exact value $\delta_2=1/9$ follows directly.

math.CO↗

arXiv2026

On the Complexity of Hop Domination and 2-Step Domination in Graph Classes

The domination problem is a well-studied problem in graph theory. In this paper, we study two natural variants: the hop domination problem and the $2$-step domination problem. Let $G$ be a graph with vertex set $V$ and edge set $E$. For a graph $G$, a subset $S \subseteq V(G)$ is called an \emph{hop dominating set} if every vertex not in $S$ lies at distance of exactly $2$ from at least one vertex in $S$. For $v\in V(G)$, let $N(v,2)$ denote the set of vertices in $V(G)$ that are at distance exactly $2$ from $v$. For a graph $G$, a subset $S \subseteq V(G)$ is called an \emph{$2$-step dominating set} if every vertex $v\in V(G)$ lies at a distance of exactly $2$ from at least one vertex in $S$. The \textsc{Hop Domination} (HD) problem and the \textsc{$2$-Step Domination} ($2$SD) problems ask whether a graph contains a hop domination set or a $2$-step domination set of size at most $k$, respectively. We study the computational complexity of these problems, and show that both are NP-complete, even when restricted to $d$-regular graphs for every $d\geq 3$, claw-free graphs and also unit disk graphs.

cs.DM↗

arXiv2026

Parameterized complexity of $r$-Hop, $r$-Step, and $r$-Hop Roman Domination

The \textsc{Dominating Set} problem is a classical and extensively studied topic in graph theory and theoretical computer science. In this paper, we examine the algorithmic complexity of several well-known exact-distance variants of domination, namely \textsc{$r$-Step Domination}, \textsc{$r$-Hop Domination}, and \textsc{$r$-Hop Roman Domination}. Let $G$ be a graph and let $r \geq 2$ be an integer. A set $S \subseteq V(G)$ is an \emph{$r$-hop dominating set} if every vertex in $V(G)\setminus S$ is at distance exactly $r$ from some vertex of $S$. Similarly, $S$ is an \emph{$r$-step dominating set} if every vertex of $G$ lies at distance exactly $r$ from at least one vertex of $S$. An \emph{$r$-hop Roman dominating function} on $G$ is a function $f \colon V(G)\to\{0,1,2\}$ such that for every vertex $v$ with $f(v)=0$, there exists a vertex $u$ at distance exactly $r$ from $v$ with $f(u)=2$. The \emph{weight} of $f$ is defined as $f(V)=\sum_{v\in V(G)} f(v)$. The \textsc{$r$-Hop Domination} (respectively, \textsc{$r$-Step Domination}) problem asks whether $G$ admits an $r$-hop dominating set (respectively, $r$-step dominating set) of size at most $k$, while the \textsc{$r$-Hop Roman Domination} problem asks whether $G$ admits an $r$-hop Roman dominating function of weight at most $k$. It is known that for every $r\ge 2$, the problems \textsc{$r$-Step Domination}, \textsc{$r$-Hop Domination}, and \textsc{$r$-Hop Roman Domination} are \textsc{NP}-complete. First we prove that for all $r\ge 2$, \textsc{$r$-Hop Roman Domination} is \textsc{W[2]}-complete. Furthermore, for every $r\ge 2$, \textsc{$r$-Step Domination} and \textsc{$r$-Hop Domination} remain \textsc{W[2]}-hard even when restricted to bipartite graphs and chordal graphs. Unless the ETH fails, none of these problems admits an algorithm running in time $2^{o(n+m)}$ on graphs with $n$ vertices and $m$ edges.

math.CO↗

arXiv2025

Exact Algorithms for Resource Reallocation Under Budgetary Constraints

Efficient resource (re-)allocation is a critical challenge in optimizing productivity and sustainability within multi-party supply networks. In this work, we introduce the \textsc{Red-Blue Reinforcement} (R-BR) problem, where a service provider under budgetary constraints must minimize client reallocations to reduce the required number of servers they should maintain by a specified amount. We conduct a systematic algorithmic study, providing three exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our algorithms are efficient for topologies that model rural road networks (bounded distance to cluster), modern transportation systems (bounded modular-width), or have bounded clique-width, a parameter that is of great theoretical importance.

cs.DS↗

arXiv2025

Minimal degenerations of orbits of skew-symmetric matrix pencils

Complete eigenstructure, e.g., eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes can be investigated qualitatively by constructing the stratification (closure hierarchy) graphs of the congruence orbits of the pencils. The results of this paper facilitate the construction of such graphs by providing all closest neighbours for a given node in the graph. More precisely, we prove a necessary and sufficient condition for one congruence orbit of a skew-symmetric matrix pencil, A, to belong to the closure of the congruence orbit of another pencil, B, such that there is no pencil, C, whose orbit contains the closure of the orbit of A and is contained in the closure of the orbit of B.

math.RT↗

arXiv2025

Canonical forms for pairs of matrices associated with Lagrangian and Dirac subspaces

We derive the canonical forms for a pair of $n\times n$ complex matrices $(E,Q)$ under transformations $(E,Q) \rightarrow (UEV,U^{-T}QV)$, and $(E,Q) \rightarrow (UEV,U^{-*}QV)$, where $U$ and $V$ are nonsingular complex matrices. We, in particular, consider the special cases of $E^TQ$ and $E^*Q$ being (skew-)symmetric and (skew-)Hermitian, respectively, that are associated with Lagrangian and Dirac subspaces and related linear-time invariant dissipative Hamiltonian descriptor systems.

math.RT↗