arXiv subjects

Dipayan Chakraborty

Publications and source records attributed to Dipayan Chakraborty.

At least 19 recordsLinked to original sources

arXiv2026

Pure State Transformations under Block Coherence

Block coherence provides a natural generalization of standard quantum coherence by treating superpositions across different subspaces as a resource. This work studies deterministic pure-state conversion under three free operations: physically block incoherent operations (PBIO), strictly block incoherent operations (SBIO), and block dephasing covariant incoherent operations (BDCO). For PBIO, we prove that, under a natural nondegeneracy condition on the active Kraus branches, any deterministic conversion from one pure state to another must be implemented by a block incoherent unitary. When the nondegeneracy requirement is removed, the condition becomes more general. It demands that the blockwise action of every active branch reproduce the target block structure with a common proportionality factor across all output blocks. For SBIO and BDCO, we show that deterministic pure-state transformation is completely characterized by the majorization relation between the input and output block probability vectors. The converse proof is constructive, yielding an explicit Kraus representation for every admissible BDCO transformation. In the rank-one limit, these conditions reduce to the known pure-state transformation criteria for physically incoherent operations (PIO), strictly incoherent operations (SIO), and dephasing covariant incoherent operations (DIO) in the standard resource theory of coherence. Using the majorization condition, a maximally block-coherent state with uniform block weights is also identified as a universal pure-state resource under BDCO and SBIO. We have also provided geometric numerical illustrations comparing the state transformation power of BDCO and DIO for a fixed input state, fixed output state and mutual convertibility scenarios.

quant-ph↗

arXiv2026

Catalytic Enhancement of Coherence in Noisy Quantum Channels and Characterization of Strictly Incoherent Operations

In realistic quantum information processing tasks, quantum states are inevitably affected by environmental noise, leading to decoherence and degradation of useful quantum resources. The coherence fraction, which serves as an important figure of merit for several quantum protocols, may decrease significantly after the action of a noisy channel. Such degradation can result in unsatisfactory performance in real-world applications. In this work, we investigate whether catalysis can be used to pre-process the input state to enhance the coherence fraction of an output state from a quantum channel. Specifically, we study whether using a processed state $\rho_s'$ as the input to a quantum channel $\Lambda$, instead of the original state $\rho_s$, can yield an output state $\Lambda(\rho_s')$ whose coherence fraction exceeds that of $\Lambda(\rho_s)$. We analyze the conditions under which such an improvement is possible. We also provide a practical application of our setup for the phase discrimination task. Furthermore, we establish a necessary and sufficient condition for an incoherent state preserving CPTP(Completely Positive Trace Preserving) map $\mathcal{E}$ to be a particular type of Strictly Incoherent Operation (SIO). This characterization provides a new structural understanding of SIO and clarifies its role in coherence manipulation. Our results offer practical insights into coherence preservation and enhancement in noisy quantum processes and may be useful for optimizing quantum information protocols under realistic conditions. We also provide numerical examples to support our claims.

quant-ph↗

arXiv2026

On the Complexity of Vertex-Splitting Into an Interval Graph

Vertex splitting is a graph modification operation in which a vertex is replaced by multiple vertices such that the union of their neighborhoods equals the neighborhood of the original vertex. We introduce and study vertex splitting as a graph modification operation for transforming graphs into interval graphs. Given a graph $G$ and an integer $k$, we consider the problem of deciding whether $G$ can be transformed into an interval graph using at most $k$ vertex splits. We prove that this problem is NP-hard, even when the input is restricted to subcubic planar bipartite graphs. We further observe that vertex splitting differs fundamentally from vertex and edge deletions as graph modification operations when the objective is to obtain a chordal graph, even for graphs with maximum independent set size at most two. On the positive side, we give a polynomial-time algorithm for transforming, via a minimum number of vertex splits, a given graph into a disjoint union of paths, and that splitting triangle free graphs into unit interval graphs is also solvable in polynomial time.

cs.CC↗

arXiv2026

The Interplay Between Domination and Separation in Graphs

In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a code of the graph. In this paper, we study the four separation properties location, closed-separation, open-separation and full-separation. We address the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes (establishing a particularly close relation between separation and codes based on domination) as well as the interplay of separation and complementation (showing that location and full-separation are the same on a graph and its complement, whereas closed-separation in a graph corresponds to open-separation in its complement).

math.CO↗

arXiv2025

Study of low-energy electron-induced dissociation of 1-Propanol

The fragmentation of 1-propanol resulting from dissociative electron attachment has been explored across an energy range of 3.5 to 16 eV. Four distinct ion species are identified: $\text{H}^{-}$, $\text{O}^{-}$, $\text{OH}^{-}$, and $\text{C}_{3}\text{H}_{7}\text{O}^{-}$. The $\text{OH}^{-}$ ion exhibited a prominent peak near 8.7 eV, along with a small hump near 5.6 eV. Complementary channels led to the formation of the $\text{H}^{-}$ and $\text{C}_{3}\text{H}_{7}\text{O}^{-}$ ions. Both these two ions exhibit a sharp peak near 6 eV and broad overlapping resonances between 7 to 12 eV. The observed ion yields of distinct dissociation fragments in this study, when compared with those from previously studied alcohols, suggest site-specific fragmentation of alcohols during dissociative electron attachment. To gain a deeper understanding of the dissociation pathways, Density Functional Theory~(DFT) calculations were conducted, revealing the threshold energies for each channel. These threshold energies aligned well with the experimental uncertainties.

physics.atm-clus↗

arXiv2024

On lower bounds for cardinalities of several separating-dominating codes in graphs

In the literature, several different identification problems in graphs have been studied, the most widely studied such problems are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a "code" of the graph. The problems of determining such codes of minimum cardinality are all shown to be NP-hard. A typical line to attack such problems is, therefore, to provide bounds on the cardinalities of the studied codes. In this paper, we are interested in extremal graphs for lower bounds in terms of the order of the graph. For some codes, logarithmic lower bounds are known from the literature. We provide for eight different identification problems a general construction of extremal graphs minimizing the cardinality of a minimum code compared to the order of the graph. This enables us to reprove the existing logarithmic lower bounds, to establish such bounds for further codes, and to characterize all graphs attaining these bounds.

math.CO↗

arXiv2024

Structural Parameterization of Locating-Dominating Set and Test Cover

We investigate structural parameterizations of two identification problems: LOCATING-DOMINATING SET and TEST COVER. In the first problem, an input is a graph $G$ on $n$ vertices and an integer $k$, and one asks if there is a subset $S$ of $k$ vertices such that any two distinct vertices not in $S$ are dominated by distinct subsets of $S$. In the second problem, an input is a set of items $U$, a set of subsets $\mathcal{F}$ of $U$ called $tests$ and an integer $k$, and one asks if there is a set $S$ of at most $k$ tests such that any two items belong to distinct subsets of tests of $S$. These two problems are "identification" analogues of DOMINATING SET and SET COVER, respectively. Chakraborty et al. [ISAAC 2024] proved that both the problems admit conditional double-exponential lower bounds and matching algorithms when parameterized by treewidth of the input graph. We continue this line of investigation and consider parameters larger than treewidth, like vertex cover number and feedback edge set number. We design a nontrivial dynamic programming scheme to solve TEST COVER in "slightly super-exponential" time $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ in the number $|U|$ of items and LOCATING-DOMINATING SET in time $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$, where $\textsf{vc}$ is the vertex cover number and $n$ is the order of the graph. This shows that the lower bound results with respect to treewidth from Chakraborty et al. [ISAAC 2024] cannot be extended to vertex cover number. We also show that, parameterized by feedback edge set number, LOCATING-DOMINATING SET admits a linear kernel thereby answering an open question in [Cappelle et al., LAGOS 2021]. Finally, we show that neither LOCATING-DOMINATING SET nor TEST COVER is likely to admit a compression algorithm returning an input with a subquadratic number of bits, unless $\textsf{NP} \subseteq \textsf{coNP}/poly$.

cs.DS↗

arXiv2024

Multipartite Monogamy of Entanglement for Three Qubit States

The distribution of entanglement in a multiparty system can be described through the principles of monogamy or polygamy. Monogamy is a fundamental characteristic of entanglement that restricts its distribution among several number of parties(more than two). In this work, our aim is to explore how quantum entanglement can be distributed in accordance with monogamy relations by utilizing both the genuine multipartite entanglement measures and bipartite entanglement measures. Specifically, we treat source entanglement as the genuine multipartite entanglement measure and use the entanglement of formation specifically for bipartite cases. For GHZ class states, we analytically demonstrate that the square of the source entanglement serves as an upper bound for the sum of the squares of the entanglement of formation of the reduced subsystems, with some exceptions for specific non-generic GHZ states. We also present numerical evidence supporting this result for W class states. Additionally, we explore the monogamy relation by using accessible entanglement as an upper bound.

quant-ph↗

arXiv2024

Progress towards the two-thirds conjecture on locating-total dominating sets

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $γ^L_t(G)$. It has been conjectured that $γ^L_t(G)\leq\frac{2n}{3}$ holds for every twin-free graph $G$ of order $n$ without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.

math.CO↗

arXiv2024

On locating and neighbor-locating colorings of sparse graphs

A proper $k$-coloring of a graph $G$ is a \emph{neighbor-locating $k$-coloring} if for each pair of vertices in the same color class, the two sets of colors found in their respective neighborhoods are different. The \textit{neighbor-locating chromatic number} $χ_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-vertex-coloring of a graph $G$ is a \emph{locating $k$-coloring} if for each pair of vertices $x$ and $y$ in the same color-class, there exists a color class $S_i$ such that $d(x,S_i)\neq d(y,S_i)$. The locating chromatic number $χ_{L}(G)$ is the minimum $k$ for which $G$ admits a locating $k$-coloring. Our main results concern the largest possible order of a sparse graph of given neighbor-locating chromatic number. More precisely, we prove that if $G$ has order $n$, neighbor-locating chromatic number $k$ and average degree at most $2a$, where $2a\le k-1$ is a positive integer, then $n$ is upper-bounded by $\mathcal{O}(a^2(k^{2a+1}))$. We also design a family of graphs of bounded maximum degree whose order is close to reaching this upper bound. Our upper bound generalizes two previous bounds from the literature, which were obtained for graphs of bounded maximum degree and graphs of bounded cycle rank, respectively. Also, we prove that determining whether $χ_L(G)\le k$ and $χ_{NL}(G)\le k$ are NP-complete for sparse graphs: more precisely, for graphs with average degree at most 7, maximum average degree at most 20 and that are $4$-partite. We also study the possible relation between the ordinary chromatic number, the locating chromatic number and the neighbor-locating chromatic number of a graph.

math.CO↗

arXiv2024

Identifying codes in triangle-free graphs of bounded maximum degree

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a constant $c$ such that for every connected closed-twin-free graph $G$ of order $n$ and maximum degree $Δ$, the graph $G$ admits an identifying code of size at most $\left( \frac{Δ-1}Δ \right) n+c$. In [D. Chakraborty, F. Foucaud, M. A. Henning, and T. Lehtilä. Identifying codes in graphs of given maximum degree: Characterizing trees. arXiv preprint arXiv:2403.13172, 2024], we proved the conjecture for all trees. In this article, we show that the conjecture holds for all triangle-free graphs, with the same list of exceptional graphs needing $c>0$ as for trees: for $Δ\ge 3$, $c=1/3$ suffices and there is only a set of 12 trees requiring $c>0$ for $Δ=3$, and when $Δ\ge 4$ this set is reduced to the $Δ$-star only. Our proof is by induction, whose starting point is the above result for trees. Along the way, we prove a generalized version of Bondy's theorem on induced subsets [J. A. Bondy. Induced subsets. Journal of Combinatorial Theory, Series B, 1972] that we use as a tool in our proofs. We also use our main result for triangle-free graphs, to prove the upper bound $\left( \frac{Δ-1}Δ \right) n+1/Δ+4t$ for graphs that can be made triangle-free by the removal of $t$ edges.

math.CO↗

arXiv2024

On full-separating sets and related codes in graphs

A domination-based identification problem on a graph $G$ is one where the objective is to choose a subset $C$ of the vertex set of $G$ such that $C$ has both, a domination property, that is, $C$ is either a dominating or a total-dominating set of $G$, and a separation property, that is, any two distinct vertices of $G$ must have distinct closed or open neighborhoods in $C$. Such a set $C$ is often referred to as a code in the literature of identification problems. In this article, we introduce a new separation property, called full-separation, as it combines aspects of the two well-studied properties of closed- and open-separation. We study it in combination with both domination and total-domination and call the resulting codes full-separating dominating codes (or FD-codes for short) and full-separating total-dominating codes (or FTD-codes for short), respectively. Incidentally, FTD-codes have also been introduced in the literature of identification problems under the name of strongly identifying codes (or SID-codes for short) and under a differently formulated definition. In this paper, we address the conditions for the existence of FD- and FTD-codes, bounds for their size, their relation to codes of the other types and present some extremal cases for these bounds and relations. We further show that the problems of determining an FD- or an FTD-code of minimum cardinality in a graph are NP-hard. We also show that the cardinalities of minimum FD- and FTD-codes of any graph differ by at most one, but that it is NP-hard to decide whether or not they are equal for a given graph in general.

math.CO↗

arXiv2024

Identifying open codes in trees and 4-cycle-free graphs of given maximum degree

An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u) \cap S \ne N_G(v) \cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v) \cap S \ne \emptyset$ for all vertices~$v$ in $G$). Such a set exists if and only if the graph $G$ is open twin-free and isolate-free; and the minimum cardinality of an identifying open code in an open twin-free and isolate-free graph $G$ is denoted by $γ^{\rm {\small IOC}}(G)$. We study the smallest size of an identifying open code of a graph, in relation with its order and its maximum degree. For $Δ$ a fixed integer at least $3$, if $G$ is a connected graph of order $n \ge 5$ that contains no $4$-cycle and is open twin-free with maximum degree bounded above by $Δ$, then we show that $γ^{\rm {\small IOC}}(G) \le \left( \frac{2Δ- 1}Δ \right) n$, unless $G$ is obtained from a star $K_{1,Δ}$ by subdividing every edge exactly once. Moreover, we show that the bound is best possible by constructing graphs that reach the bound.

math.CO↗

arXiv2024

The $n/2$-bound for locating-dominating sets in subcubic graphs

The location-domination number is conjectured to be at most half of the order for twin-free graphs with no isolated vertices. We prove that this conjecture holds and is tight for subcubic graphs. We also show that the same upper bound holds for subcubic graphs with open twins of degree 3 and closed twins of any degree, but not for subcubic graphs with open twins of degree 1 or 2. These results then imply that the same upper bound holds for all cubic graphs (with or without twins) except $K_4$ and $K_{3,3}$.

math.CO↗

arXiv2024

A linear algorithm for radio $k$-coloring of powers of paths having small diameters

The radio $k$-chromatic number $rc_k(G)$ of a graph $G$ is the minimum integer $λ$ such that there exists a function $ϕ: V(G) \to \{0,1,\cdots, λ\}$ satisfying $|ϕ(u)-ϕ(v)| \geq k+1 - d(u,v)$, where $d(u,v)$ denotes the distance between $u$ and $v$. A considerable amount of attention has been given to find the exact values or providing polynomial time algorithms to determine $rc_k(G)$ for several basic graph families such as paths, cycles, trees, and powers of paths, usually for some specific values of $k$. In this article, we find the exact values of $rc_k(G)$ where $G$ is a power of a path with diameter strictly less than $k$. Our proof readily provides a linear time algorithm for assigning a radio $k$-coloring of $G$. Furthermore, our proof technique is a potential tool for solving the same problem for other classes of graphs having ``small'' diameters.

math.CO↗

arXiv2024

Identifying codes in graphs of given maximum degree: Characterizing trees

An identifying code of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists an absolute constant $c$ such that for every connected graph $G$ of order $n$ and maximum degree $\Delta$, the graph $G$ admits an identifying code of size at most $( \frac{\Delta-1}{\Delta} )n +c$. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant $c$ together with the exact value of the constant. Hence, proving the conjecture for trees. For $\Delta=2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $\Delta\ge 3$, we show that $c=1/\Delta\le 1/3$ suffices and that $c$ is required to have a positive value only for a finite number of trees. In particular, for $\Delta = 3$, there are 12 trees with a positive constant $c$ and, for each $\Delta \ge 4$, the only tree with positive constant $c$ is the $\Delta$-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtil\"a. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022]. We remark that there are infinitely many trees for which the bound is tight when $\Delta=3$; for every $\Delta\ge 4$, we construct an infinite family of trees of order $n$ with identification number very close to the bound, namely $\left( \frac{\Delta-1+\frac{1}{\Delta-2}}{\Delta+\frac{2}{\Delta-2}} \right) n > (\frac{\Delta-1}{\Delta} ) n -\frac{n}{\Delta^2}$. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree $T$ is at most its number of vertices.

math.CO↗

arXiv2024

Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove the (tight) conditional lower bounds for these problems when parameterized by treewidth and solution as. Formally, \textsc{Locating-Dominating Set} (respectively, \textsc{Test Cover}) parameterized by the treewidth of the input graph (respectively, of the natural auxiliary graph) does not admit an algorithm running in time $2^{2^{o(tw)}} \cdot poly(n)$ (respectively, $2^{2^{o(tw)}} \cdot poly(|U| + |\mathcal{F}|))$. This result augments the small list of NP-Complete problems that admit double exponential lower bounds when parameterized by treewidth. Then, we first prove that \textsc{Locating-Dominating Set} does not admit an algorithm running in time $2^{o(k^2)} \cdot poly(n)$, nor a polynomial time kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(k)}$ vertices, unless the \ETH\ fails. To the best of our knowledge, \textsc{Locating-Dominating Set} is the first problem that admits such an algorithmic lower-bound (with a quadratic function in the exponent) when parameterized by the solution size. Finally, we prove that \textsc{Test Cover} does not admit an algorithm running in time $2^{2^{o(k)}} \cdot poly(|U| + |\mathcal{F}|)$. This is also a rare example of the problem that admits a double exponential lower bound when parameterized by the solution size. We also present algorithms whose running times match the above lower bounds.

cs.DS↗

arXiv2024

On open-separating dominating codes in graphs

Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set $C$ of a graph $G$ which is also separating in the sense that the neighbourhoods of any two distinct vertices of $G$ have distinct intersections with $C$. Such a dominating and separating set $C$ of a graph is often referred to as a code in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called open-separating dominating code, or OD-code for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OD-codes. Due to the emergence of a close and yet difficult to establish relation of the OD-code with another well-studied code in the literature called open (neighborhood)-locating dominating code (referred to as the open-separating total-dominating code and abbreviated as OTD-code in this paper), we compare the two codes on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding OD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OD-codes, again in relation to OTD-codes of some graph families already studied in this context.

math.CO↗