arXiv subjects

Sandip Das

Publications and source records attributed to Sandip Das.

At least 19 recordsLinked to original sources

arXiv2026

An Improved Upper Bound for the Tur\'an Number of the Hexagon

For a graph $F$, the Tur\'an number $\operatorname{ex}(n,F)$ is the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Determining the Tur\'an numbers of even cycles is a central problem in extremal graph theory and remains open in general. For $C_6$, the best previous upper bound was due to F\"uredi, Naor, and Verstra\"ete [Advances in Mathematics, 2006], who proved that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq \lambda n^{4/3}+O(n)<0.6272 n^{4/3}, $$ where $\lambda$ is the real root of $ 16\lambda^3-4\lambda^2+\lambda-3=0$. We improve this bound by showing that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq \alpha n^{4/3}+O(n)<0.6144 n^{4/3}, $$ where $\alpha$ is the unique real root of $ 4 \alpha^{3} (3/2)^{1-1/(2\alpha)} =1$ in the interval $(1/2,2/3)$.

math.CO↗

arXiv2026

Minimum eccentricity shortest paths of $K_{2,3}$-minor-free graphs

Given a simple, undirected, and unweighted graph $G$, and an integer $R$, the objective of the \textsc{Minimum Eccentricity Shortest Path (MESP)} is to decide whether there exists an \emph{isometric path} $P$ in $G$ such that the distance from every vertex in the graph to its nearest vertex in $P$ is at most $R$. In this paper, we prove that MESP admits an $O(n^4)$-time algorithm on $K_{2,3}$-minor-free graphs. Our algorithm has a cubic running time when the inputs are restricted to a cactus.

cs.DS↗

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

Algorithms and complexity for geodetic sets on interval and chordal graphs

We study the computational complexity of finding the geodetic number of a graph on chordal graphs and interval graphs. A set $S$ of vertices of a graph $G$ is a \textit{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. We show that \textsc{Minimum Geodetic Set} is fixed parameter tractable for chordal graphs when parameterized by its \emph{tree-width} (which equals its clique number). This implies a polynomial-time algorithm for $k$-trees, for fixed $k$. Then, we show that \textsc{Minimum Geodetic Set} is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012), who showed that \textsc{Minimum Geodetic Set} is polynomial-time solvable on proper interval graphs. As interval graphs are very constrained, to prove the latter result, we design a rather sophisticated reduction technique to work around their inherent linear structure.

cs.DS↗

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

Nonlinear Energy Transfer Analysis between Coherent Modes in Developing Plasma Turbulence

Energy transfer among various spectral components of fluctuating physical parameters in plasma occurs due to the nonlinear interactions, but these effects are typically not captured by the traditional linear spectral methods. Plasma density fluctuations measured in the Inverse Mirror Plasma Experimental Device (IMPED) have signatures of nonlinear mode interactions among various instability modes, i.e. Rayleigh-Taylor (RT) and Drift-Wave (DW) modes. In this paper, the energy transfer among these modes as a result of nonlinear wave interactions (through the quadratic coupling processes) have been investigated in detail. The existing computational methods for single field model such as Ritz method and Kim method have been explored to understand the dynamics of nonlinear interaction between coherent modes. Both methods are applied and validated in simulation as well as experimental data from IMPED for coherent modes in plasma. We find that the validity and applicability of the methods depend on the statistical nature of the data, particularly higher-order moments such as kurtosis, and on spatial stationarity. Energy transfer analysis using these methods reveals primarily the transfer of energy from a RT mode and a mix mode of DW and RT to a comparatively low-frequency DW mode, demonstrating the capability of the method to quantify spectral energy transport between the coherent modes.

physics.plasm-ph↗

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↗

arXiv2026

On the complexity of Multipacking

A multipacking in an undirected graph $G=(V,E)$ is a set $M\subseteq V$ such that for every vertex $v\in V$ and for every integer $r\geq 1$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The Multipacking problem asks whether a graph contains a multipacking of size at least $k$. For more than a decade, it remained an open question whether the Multipacking problem is NP-complete or solvable in polynomial time. Whereas the problem is known to be polynomial-time solvable for certain graph classes (e.g., strongly chordal graphs, grids, etc). Foucaud, Gras, Perez, and Sikora [Algorithmica 2021] made a step towards solving the open question by showing that the Multipacking problem is NP-complete for directed graphs and it is W[1]-hard when parameterized by the solution size. In this paper, we prove that the Multipacking problem is NP-complete for undirected graphs, which answers the open question. Moreover, the problem is W[2]-hard for undirected graphs when parameterized by the solution size. Furthermore, we have shown that the problem is NP-complete and W[2]-hard (when parameterized by the solution size) even for various subclasses: chordal, bipartite, and claw-free graphs. Whereas, it is NP-complete for regular, and CONV graphs (intersection graphs of convex sets in the plane). Additionally, the problem is NP-complete and W[2]-hard (when parameterized by the solution size) for chordal $\cap$ $\frac{1}{2}$-hyperbolic graphs, which is a superclass of strongly chordal graphs where the problem is polynomial-time solvable. On the positive side, we present an exact exponential-time algorithm for the Multipacking problem on $n$-vertex general graphs, which breaks the $2^n$ barrier by achieving a running time of $O^*(1.58^n)$.

cs.CC↗

arXiv2026

On the Number of Almost Empty Monochromatic Triangles

In this paper, we consider the problem of counting almost empty monochromatic triangles in colored planar point sets, that is, triangles whose vertices are all assigned the same color and that contain only a few interior points. Specifically, we show that any $c$-coloring of a set of $n$ points in the plane in general position (that is, no three on a line) contains $\Omega(n^2)$ monochromatic triangles with at most $c-1$ interior points and $\Omega(n^{\frac{4}{3}})$ monochromatic triangles with at most $c-2$ interior points, for any fixed $c \geq 2$. The latter, in particular, generalizes the result of Pach and T\'{o}th (2013) on the number of monochromatic empty triangles in 2-colored point sets, to the setting of multiple colors and monochromatic triangles with a few interior points. We also derive the limiting value of the expected number of triangles with $s$ interior points in random point sets, for any integer $s \geq 0$. As a result, we obtain the expected number of monochromatic triangles with at most $s$ interior points in random colorings of random point sets.

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↗

arXiv2024

Multipacking in Euclidean Metric Space

Here we study the multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v \in P$ and the point $v$ itself. We assume that the \emph{$s$-th neighbor} of each point is unique, for every $s \in \{0, 1, 2, \dots , n-1\}$. For a natural number $r \leq n-1$, an $r$-multipacking is a set $ M \subseteq P $ such that for each point $ v \in P $ and for every integer $ 1\leq s \leq r $, $|N_s[v]\cap M|\leq (s+1)/2$. The $r$-multipacking number of $ P $ is the maximum cardinality of an $r$-multipacking of $ P $ and is denoted by $ \MP_{r}(P) $. For $r=n-1$, an $r$-multipacking is called a multipacking and $r$-multipacking number is called as multipacking number. For $r=1 \text{ and } 2$, we study the problem of computing a maximum $r$-multipacking of the point sets in $\mathbb{R}^2$. We show that a maximum $1$-multipacking can be computed in polynomial time but computing a maximum $2$-multipacking is \textsc{NP-hard}. Further, we provide approximation and parameterized solutions to the $2$-multipacking problem.

cs.CG↗

arXiv2024

On the Cop Number of String Graphs

Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The \emph{cop number} of a graph is the minimum number of cops required to capture the robber. Gaven\v{c}iak et al.~[Eur. J. of Comb. 72, 45--69 (2018)] studied the game on intersection graphs and established that the cop number for the class of string graphs is at most 15, and asked as an open question to improve this bound for string graphs and subclasses of string graphs. We address this question and establish that the cop number of a string graph is at most 13. To this end, we develop a novel \textit{guarding} technique. We further establish that this technique can be useful for other Cops and Robber games on graphs admitting a representation. In particular, we show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al.~[Austr. J. Comb. 76(2), 248--265 (2020)]. In passing, we also improve the known bounds on the cop number of boxicity 2 graphs. Finally, as a corollary of our result on the cop number of string graphs, we establish that the chromatic number of string graphs with girth at least $5$ is at most $14$.

cs.DM↗

arXiv2024

Growth Rate of the Number of Empty Triangles in the Plane

Given a set $P$ of $n$ points in the plane, in general position, denote by $N_\Delta(P)$ the number of empty triangles with vertices in $P$. In this paper we investigate by how much $N_\Delta(P)$ changes if a point $x$ is removed from $P$. By constructing a graph $G_P(x)$ based on the arrangement of the empty triangles incident on $x$, we transform this geometric problem to the problem of counting triangles in the graph $G_P(x)$. We study properties of the graph $G_P(x)$ and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemer\'edi problem.

cs.DM↗

arXiv2023

Relation between broadcast domination and multipacking numbers on chordal and other hyperbolic graphs

For a graph $ G = (V, E) $ with a vertex set $ V $ and an edge set $ E $, a function $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each vertex $ u \in V $, if there exists a vertex $ v $ in $ G $ (possibly, $ u = v $) such that $ f (v) > 0 $ and $ d(u, v) \leq f (v) $, then $ f $ is called a dominating broadcast on $ G $. The cost of the dominating broadcast $f$ is the quantity $ \sum_{v\in V}f(v) $. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $ \gamma_{b}(G) $. A multipacking is a set $ S \subseteq V $ in a graph $ G = (V, E) $ such that for every vertex $ v \in V $ and for every integer $ r \geq 1 $, the ball of radius $ r $ around $ v $ contains at most $ r $ vertices of $ S $, that is, there are at most $ r $ vertices in $ S $ at a distance at most $ r $ from $ v $ in $ G $. The multipacking number of $ G $ is the maximum cardinality of a multipacking of $ G $ and is denoted by $ mp(G) $. We show that, for any connected chordal graph $G$, $\gamma_{b}(G)\leq \big\lceil{\frac{3}{2} mp(G)\big\rceil}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio $\gamma_b(G)/mp(G)=10/9$, with $mp(G)$ arbitrarily large. Moreover, we show that $\gamma_{b}(G)\leq \big\lfloor{\frac{3}{2} mp(G)+2\delta\big\rfloor} $ holds for all $\delta$-hyperbolic graphs. In addition, we provide a polynomial-time algorithm to construct a multipacking of a $\delta$-hyperbolic graph $G$ of size at least $ \big\lceil{\frac{2mp(G)-4\delta}{3} \big\rceil} $.

cs.DM↗

arXiv2023

Multipacking and broadcast domination on cactus graph and its impact on hyperbolic graph

For a graph $G$, $ mp(G) $ is the multipacking number, and $\gamma_b(G)$ is the broadcast domination number. It is known that $mp(G)\leq \gamma_b(G)$ and $\gamma_b(G)\leq 2mp(G)+3$ for any graph $G$, and it was shown that $\gamma_b(G)-mp(G)$ can be arbitrarily large for connected graphs. It is conjectured that $\gamma_b(G)\leq 2mp(G)$ for any general graph $G$. We show that, for any cactus graph $G$, $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for cactus graphs and asteroidal triple-free graphs by constructing an infinite family of cactus graphs which are also asteroidal triple-free graphs such that the ratio $\gamma_b(G)/mp(G)=4/3$, with $mp(G)$ arbitrarily large. This result shows that, for cactus graphs, the bound $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ cannot be improved to a bound in the form $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$, for any constant $c_1<4/3$ and $c_2$. Moreover, we provide an $O(n)$-time algorithm to construct a multipacking of cactus graph $G$ of size at least $ \frac{2}{3}mp(G)-\frac{11}{3} $, where $n$ is the number of vertices of the graph $G$. The hyperbolicity of the cactus graph class is unbounded. For $0$-hyperbolic graphs, $mp(G)=\gamma_b(G)$. Moreover, $mp(G)=\gamma_b(G)$ holds for the strongly chordal graphs which is a subclass of $\frac{1}{2}$-hyperbolic graphs. Now it's a natural question: what is the minimum value of $\delta$, for which we can say that the difference $ \gamma_{b}(G) - mp(G) $ can be arbitrarily large for $\delta$-hyperbolic graphs? We show that the minimum value of $\delta$ is $\frac{1}{2}$ using a construction of an infinite family of cactus graphs with hyperbolicity $\frac{1}{2}$.

cs.DM↗

arXiv2023

Burning a binary tree and its generalization

Graph burning is a graph process that models the spread of social contagion. Initially, all the vertices of a graph $G$ are unburnt. At each step, an unburnt vertex is put on fire and the fire from burnt vertices of the previous step spreads to their adjacent unburnt vertices. This process continues till all the vertices are burnt. The burning number $b(G)$ of the graph $G$ is the minimum number of steps required to burn all the vertices in the graph. The burning number conjecture by Bonato et al. states that for a connected graph $G$ of order $n$, its burning number $b(G) \leq \lceil \sqrt{n} \rceil$. It is easy to observe that in order to burn a graph it is enough to burn its spanning tree. Hence it suffices to prove that for any tree $T$ of order $n$, its burning number $b(T) \leq \lceil \sqrt{n} \rceil$ where $T$ is the spanning tree of $G$. It was proved in 2018 that $b(T) \leq \lceil \sqrt{n + n_2 + 1/4} +1/2 \rceil$ for a tree $T$ where $n_2$ is the number of degree $2$ vertices in $T$. In this paper, we provide an algorithm to burn a tree and we improve the existing bound using this algorithm. We prove that $b(T)\leq \lceil \sqrt{n + n_2 + 8}\rceil -1$ which is an improved bound for $n\geq 50$. We also provide an algorithm to burn some subclasses of the binary tree and prove the burning number conjecture for the same.

math.CO↗

arXiv2023

Large scale synthesis of 2D graphene oxide by mechanical milling of 3D carbon nanoparticles in air

Graphene oxide (GO) is one of the important functional materials. Large-scale synthesis of it is very challenging. Following a simple cost-effective route, large-scale GO was produced by mechanical (ball) milling, in air, of carbon nanoparticles (CNPs) present in carbon soot in the present study. The thickness of the GO layer was seen to decrease with an increase in milling time. Ball milling provided the required energy to acquire the in-plane graphitic order in the CNPs reducing the disorders in it. As the surface area of the layered structure became more and more with the increase in milling time, more and more oxygen of air got attached to the carbon in graphene leading to the formation of GO. An increase in the time of the ball mill up to 5 hours leads to a significant increase in the content of GO. Thus ball milling can be useful to produce large-scale two-dimensional GO for a short time.

cond-mat.mtrl-sci↗

arXiv2023

On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters

An $(n,m)$-graph is characterised by having $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to an $(n,m)$-graph $H$, is a vertex mapping that preserves adjacency, direction, and type. The $(n,m)$-chromatic number of $G$, denoted by $\chi_{n,m}(G)$, is the minimum value of $|V(H)|$ such that there exists a homomorphism of $G$ to $H$. The theory of homomorphisms of $(n,m)$-graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of $G$ is bounded by a function of $\chi_{n,m}(G)$ but not the other way around. Additionally, we show that the acyclic chromatic number of $G$ is bounded by a function of $\chi_{n,m}(G)$, a result already known in the reverse direction. Furthermore, we prove that the $(n,m)$-chromatic number for the family of graphs with a maximum average degree less than $2+ \frac{2}{4(2n+m)-1}$, including the subfamily of planar graphs with girth at least $8(2n+m)$, equals $2(2n+m)+1$. This improves upon previous findings, which proved the $(n,m)$-chromatic number for planar graphs with girth at least $10(2n+m)-4$ is $2(2n+m)+1$. It is established that the $(n,m)$-chromatic number for the family $\mathcal{T}_2$ of partial $2$-trees is both bounded below and above by quadratic functions of $(2n+m)$, with the lower bound being tight when $(2n+m)=2$. We prove $14 \leq \chi_{(0,3)}(\mathcal{T}_2) \leq 15$ and $14 \leq \chi_{(1,1)}(\mathcal{T}_2) \leq 21$ which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.

math.CO↗