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Debojyoti Bhattacharya

Publications and source records attributed to Debojyoti Bhattacharya.

9 recordsLinked to original sources
arXiv2026

Ulrich bundles on smooth toric threefolds with Picard number $2$

In this paper, we study Ulrich bundles on smooth toric threefolds with Picard number$~2$, namely $\mathbb P(\mathcal O_{\mathbb P^{2}}(a_0) \oplus \mathcal O_{\mathbb P^{2}}(a_1))$. We construct resolutions and monads for Ulrich bundles of arbitrary rank, and provide explicit examples together with a complete classification of those arising as pullbacks from $\mathbb{P}^2$. As a consequence, we also show that these varieties are Ulrich wild.

math.AG↗
arXiv2025

Liar's vertex-edge domination in unit disk graph

Let $G=(V, E)$ be a simple undirected graph. A closed neighbourhood of an edge $e=uv$ between two vertices $u$ and $v$ of $G$, denoted by $N_G[e]$, is the set of vertices in the neighbourhood of $u$ and $v$ including $\{u,v\}$. A subset $L$ of $V$ is said to be liar's vertex-edge dominating set if $(i)$ for every edge $e\in E$, $|N_G[e]\cap L|\geq 2$ and $(ii)$ for every pair of distinct edges $e,e'$, $|(N_G[e]\cup N_G[e'])\cap L|\geq 3$. The minimum liar's vertex-edge domination problem is to find the liar's vertex-edge dominating set of minimum cardinality. In this article, we show that the liar's vertex-edge domination problem is NP-complete in unit disk graphs, and we design a polynomial time approximation scheme(PTAS) for the minimum liar's vertex-edge domination problem in unit disk graphs.

math.CO↗
arXiv2025

Liar's vertex-edge domination in subclasses of chordal graphs

Let $G=(V, E)$ be an undirected graph. The set $N_G[x]=\{y\in V|xy\in E\}\cup \{x\}$ is called the closed neighbourhood of a vertex $x\in V$ and for an edge $e=xy\in E$, the closed neighbourhood of $e$ is the set $N_G[x]\cup N_G[y]$, which is denoted by $N_G[e]$ or $N_G[xy]$. A set $L\subseteq V$ is called \emph{liar's vertex-edge dominating set} of a graph $G=(V,E)$ if for every $e_i\in E$, $|N_G[e_i]\cap L|\geq 2$ and for every pair of distinct edges $e_i,e_j\in E$, $|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$. The notion of liar's vertex-edge domination arises naturally from some applications in communication networks. Given a graph $G$, the \textsc{Minimum Liar's Vertex-Edge Domination Problem} (\textsc{MinLVEDP}) asks to find a liar's vertex-edge dominating set of $G$ of minimum cardinality. In this paper, we study this problem from an algorithmic point of view. We design two linear time algorithms for \textsc{MinLVEDP} in block graphs and proper interval graphs, respectively. On the negative side, we show that the decision version of liar's vertex-edge domination problem is NP-complete for undirected path graphs.

math.CO↗
arXiv2024

$\ell$-away ACM line bundles on a nonsingular cubic surface

Let $X \subset \mathbb P^3$ be a nonsingular cubic hypersurface. Faenzi (\cite{F}) and later Pons-Llopis and Tonini (\cite{PLT}) have completely characterized ACM line bundles over $X$. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify $\ell$-away ACM line bundles (introduced recently by Gawron and Genc (\cite{GG})) over $X$, when $\ell \leq 2$. For $\ell\geq 3$, we give examples of $\ell$-away ACM line bundles on $X$ and for each $\ell \geq 1$, we establish the existence of smooth hypersurfaces $X^{(d)}$ of degree $d >\ell$ in $\mathbb P^3$ admitting $\ell$-away ACM line bundles.

math.AG↗
arXiv2024

Algorithmic study on liar's vertex-edge domination problem

Let $G=(V,E)$ be a graph. For an edge $e=xy\in E$, the closed neighbourhood of $e$, denoted by $N_G[e]$ or $N_G[xy]$, is the set $N_G[x]\cup N_G[y]$. A vertex set $L\subseteq V$ is liar's vertex-edge dominating set of a graph $G=(V,E)$ if for every $e_i\in E$, $|N_G[e_i]\cap L|\geq 2$ and for every pair of distinct edges $e_i$ and $e_j$, $|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$. This paper introduces the notion of liar's vertex-edge domination which arises naturally from some applications in communication networks. Given a graph $G$, the \textsc{Minimum Liar's Vertex-Edge Domination Problem} (\textsc{MinLVEDP}) asks to find a liar's vertex-edge dominating set of $G$ of minimum cardinality. In this paper, we study this problem from algorithmic point of view. We show that \textsc{MinLVEDP} can be solved in linear time for trees, whereas the decision version of this problem is NP-complete for chordal graphs, bipartite graphs, and $p$-claw free graphs for $p\geq 4$. We further study approximation algorithms for this problem. We propose two approximation algorithms for \textsc{MinLVEDP} in general graphs and $p$-claw free graphs. %We propose an $O(\ln Δ(G))$-approximation algorithm for \textsc{MinLVEDP} in general graphs, where $Δ(G)$ is the maximum degree of the input graph. Also, we design a constant factor approximation algorithm for $p$-claw free graphs. On the negative side, we show that the \textsc{MinLVEDP} cannot be approximated within $\frac{1}{2}(\frac{1}{8}-ε)\ln|V|$ for any $ε>0$, unless $NP\subseteq DTIME(|V|^{O(\log(\log|V|)})$. Finally, we prove that the \textsc{MinLVEDP} is APX-complete for bounded degree graphs and $p$-claw free graphs for $p\geq 6$.

cs.DS↗
arXiv2023

On $k$-vertex-edge domination of graph

Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $N_G(v)=\{u\in V~|~ uv\in E\}$; whereas the closed neighbourhood is defined as $N_G[v]= N_G(v)\cup \{v\}$. For an integer $k$, a subset $D\subseteq V$ is called a $k$-vertex-edge dominating set of $G$ if for every edge $uv\in E$, $|(N_G[u]\cup N_G[v]) \cap D|\geq k$. In $k$-vertex-edge domination problem, our goal is to find a $k$-vertex-edge dominating set of minimum cardinality of an input graph $G$. In this paper, we first prove that the decision version of $k$-vertex-edge domination problem is NP-complete for chordal graphs. On the positive side, we design a linear time algorithm for finding a minimum $k$-vertex-edge dominating set of tree. We also prove that there is a $O(\log(Δ(G)))$-approximation algorithm for this problem in general graph $G$, where $Δ(G)$ is the maximum degree of $G$. Then we show that for a graph $G$ with $n$ vertices, this problem cannot be approximated within a factor of $(1-ε) \ln n$ for any $ε>0$ unless $NP\subseteq DTIME(|V|^{O(\log\log|V|)})$. Finally, we prove that it is APX-complete for graphs with bounded degree $k+3$.

math.CO↗
arXiv2022

Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson Conjecture

Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we study the configuration of points of certain reduced zero dimensional subschemes on $S$ satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small $c_2$. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of $\mathcal{M}(H, 11)$ and prove the conjecture partially. We will also show that $\mathcal{M}(H, c_2)$ is irreducible for $c_2 \le 10$ .

math.AG↗
arXiv2021

Geometry of certain Brill-Noether locus on a very general sextic surface and Ulrich bundles

Let $X \subset \mathbb P^3$ be a very general sextic surface over complex numbers. In this paper we study certain Brill-Noether problems for moduli of rank $2$ stable bundles on $X$ and its relation with rank $2$ weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill-Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank $2$ weakly Ulrich bundles $\mathcal E$ on $X$ with $c_1(\mathcal E) =5H$ and $c_2 \geq 91$ and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill-Noether loci. Finally, using the existence of rank $2$ Ulrich bundles on $X$ we obtain some more non-empty Brill-Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on $X$.

math.AG↗