arXiv subjects

Subhabrata Paul

Publications and source records attributed to Subhabrata Paul.

At least 19 recordsLinked to original sources

arXiv2026

On the Complexity and Algorithms for the Upper Domatic Number of Graphs

Let $G$ be a graph. For two disjoint vertex sets $A,B\subseteq V(G)$, we say that $A$ dominates $B$ if every vertex of $B$ has a neighbour in $A$. An upper domatic partition of $G$ is a partition $π=\{V_1,V_2,\ldots,V_k\}$ of $V(G)$ such that, for every two distinct parts $V_i$ and $V_j$, either $V_i$ dominates $V_j$, or $V_j$ dominates $V_i$, or both. The maximum order of such a partition is the upper domatic number of $G$, denoted by $D(G)$. In this paper, we study the computational complexity of the upper domatic number from both hardness and algorithmic perspectives. Motivated by the complexity questions raised in Phillips's thesis, we first prove that deciding whether $D(G)\geq k$ is NP-complete when $k$ is part of the input. Our reduction is from \textsc{Clique}, and it also shows NP-completeness for connected graphs of diameter two. On the positive side, we give exact algorithms for several graph classes. We prove that, for cographs, $D(G)=\Tr(G)$, and we provide an $O(n^3)$-time cotree dynamic program. For unicyclic graphs, we use the known equality $D(G)=\Tr(G)$. By deleting an edge of the unique cycle, we reduce the computation of $\Tr(G)$ to the transitivity of a tree and one additional decision problem testing whether adding back the deleted edge increases the transitivity by one. This yields an $O(n^3)$-time algorithm for computing the transitivity, and hence the upper domatic number, of unicyclic graphs. We further prove that $D(G)=\Tr(G)$ for complements of bipartite graphs, which yields a linear-time algorithm for complements of bipartite chain graphs. Finally, we show that $D(G)=\Tr(G)$ for split graphs and obtain a linear-time algorithm for this class.

math.CO↗

arXiv2026

Systematic Constructions of Complementary Sets and Hadamard Matrices from Circulant Operator

A Hadamard matrix $H$ of order $n$ is a square matrix with entries $\pm 1$ satisfying $HH^T = nI_n$, where $I_n$ is the identity matrix of order $n$. A circulant Hadamard matrix is a Hadamard matrix whose rows are cyclic shifts of one another. This work establishes a unified algebraic framework that treats arbitrary Hadamard matrices as flexible seeds to systematically generate Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS) through algebraic transformations. In this paper, a systematic framework using cyclic operators is presented. First, circulant Hadamard matrices of order 4 are utilized recursively to propose binary CZCS of arbitrary lengths, achieving a maximum ZCZ ratio of 2/3, and binary GCS. Significantly, this framework is generalized to establish that by employing binary or complex Hadamard matrices of any order, binary or non-binary CZCSs of arbitrary lengths can be constructed with a ZCZ ratio of 1/2. Furthermore, to provide flexible user capacity, an alternative construction of binary GCS of all lengths and Hadamard matrices of order $2^{a+1} 10^b 26^c$ ($a, b, c \geq 0$) is proposed using circulant matrices and Golay complementary pairs (GCP). These constructions are further extended to form binary CCC with parameters $(2N, 2N, 2N)$, where $N=2^a 10^b 26^c$, and $(4n, 4n, 4n)$ for $n \geq 1$. Additionally, optimal binary $(8n, 8n, 8n, 4n)$-CZCSS and their complex versions with parameters $(2m, 2m, 2m, m)$ are proposed for $n, m \geq 1$. These results provide the first generalized framework for constructing optimal CZCSS from arbitrary Hadamard seeds. Finally, a theoretical relation between Hadamard matrices and GCSs is established, and fundamental properties of circulant matrices over aperiodic correlation functions are presented.

eess.SP↗

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↗

arXiv2025

Systematic Construction of Golay Complementary Sets of Arbitrary Lengths and Alphabet Sizes

One of the important applications of Golay complementary sets (GCSs) is the reduction of peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency division multiplexing (OFDM) systems. OFDM has played a major role in modern wireless systems such as long-term-evolution (LTE), 5th generation (5G) wireless standards, etc. This paper searches for systematic constructions of GCSs of arbitrary lengths and alphabet sizes. The proposed constructions are based on extended Boolean functions (EBFs). For the first time, we can generate codes of independent parameter choices.

cs.IT↗

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

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↗

arXiv2023

Algorithmic study on $2$-transitivity of graphs

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. In this article, we study a variation of transitive partition, namely \emph{$2$-transitive partition}. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$2$-dominates} $B$ if every vertex of $B$ is adjacent to at least two vertices of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{$2$-transitive partition} of size $k$ if $V_i$ $2$-dominates $V_j$ for all $1\leq i<j\leq k$. The \textsc{Maximum $2$-Transitivity Problem} is to find a $2$-transitive partition of a given graph with the maximum number of parts. We show that the decision version of this problem is NP-complete for chordal and bipartite graphs. On the positive side, we design three linear-time algorithms for solving \textsc{Maximum $2$-Transitivity Problem} in trees, split and bipartite chain graphs.

math.CO↗

arXiv2023

Strong transitivity of a graph

A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{strongly dominates} $B$ if for every vertex $y\in B$, there exists a vertex $x\in A$, such that $xy\in E$ and $deg_G(x)\geq deg_G(y)$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{strong transitive partition} of size $k$ if $V_i$ strongly dominates $V_j$ for all $1\leq i<j\leq k$. The \textsc{Maximum Strong Transitivity Problem} is to find a strong transitive partition of a given graph with the maximum number of parts. In this article, we initiate the study of this variation of transitive partition from algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs. On the positive side, we prove that this problem can be solved in linear time for trees and split graphs.

math.CO↗

arXiv2023

Algorithmic study of $d_2$-transitivity of graphs

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. In this article, we initiate the study of a generalization of transitive partition, namely \emph{$d_2$-transitive partition}. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$d_2$-dominates} $B$ if, for every vertex of $B$, there exists a vertex in $A$, such that the distance between them is at most two. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{$d_2$-transitive partition} of size $k$ if $V_i$ $d_2$-dominates $V_j$ for all $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{$d_2$-transitivity} of $G$, and it is denoted by $Tr_{d_2}(G)$. The \textsc{Maximum $d_2$-Transitivity Problem} is to find a $d_2$-transitive partition of a given graph with the maximum number of parts. We show that this problem can be solved in linear time for the complement of bipartite graphs and bipartite chain graphs. On the negative side, we prove that the decision version of the \textsc{Maximum $d_2$-Transitivity Problem} is NP-complete for split graphs, bipartite graphs, and star-convex bipartite graphs.

math.CO↗

arXiv2023

A Direct Construction of Optimal Symmetrical Z-Complementary Code Sets of Prime Power Lengths

This paper presents a direct construction of an optimal symmetrical Z-complementary code set (SZCCS) of prime power lengths using a multi-variable function (MVF). SZCCS is a natural extension of the Z-complementary code set (ZCCS), which has only front-end zero correlation zone (ZCZ) width. SZCCS has both front-end and tail-end ZCZ width. SZCCSs are used in developing optimal training sequences for broadband generalized spatial modulation systems over frequency-selective channels because they have ZCZ width on both the front and tail ends. The construction of optimal SZCCS with large set sizes and prime power lengths is presented for the first time in this paper. Furthermore, it is worth noting that several existing works on ZCCS and SZCCS can be viewed as special cases of the proposed construction.

cs.IT↗

arXiv2022

Transitivity on subclasses of chordal graphs

Let $G=(V, E)$ be a graph, where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive partition of a given graph with the maximum number of partitions. It was known that the decision version of \textsc{Maximum Transitivity Problem} is NP-complete for chordal graphs [Iterated colorings of graphs, \emph{Discrete Mathematics}, 278, 2004]. In this paper, we first prove that this problem can be solved in linear time for \emph{split graphs} and for the \emph{complement of bipartite chain graphs}, two subclasses of chordal graphs. We also discuss Nordhaus-Gaddum type relations for transitivity and provide counterexamples for an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018]. Finally, we characterize transitively critical graphs having fixed transitivity.

math.CO↗

arXiv2022

A Direct Construction of Cross Z-Complementary Sets with Flexible Lengths and Large Zero Correlation Zone

This letter proposes a direct construction for cross Z-complementary sets (CZCSs) with flexible lengths and a large zero correlation zone (ZCZ). CZCS is an extension of the cross Z-complementary pair (CZCP). The maximum possible ZCZ width of a CZCP is half of its sequence length. In this letter, for the first time, a generalized Boolean function based construction of CZCSs with a large number of constituent sequences and a ZCZ ratio of $2/3$ is presented. For integers $m$ and $δ$, the proposed construction produces CZCS with length expressed as $2^{m-1}+2^δ$ ($0 \leq δ<m-1,m\geq 4$), where both odd and even lengths CZCS can be obtained. Additionally, the constructed CZCS also feature a complementary set of the same length. Finally, the proposed construction is compared with the existing works.

cs.IT↗

arXiv2022

A Direct Construction of Cross Z-Complementary Sequence Sets with Large Set Size

This letter presents a direct construction of cross Z-complementary sequence sets (CZCSSs), whose aperiodic correlation sums exhibit zero correlation zones at both the front-end and tail-end shifts. CZCSS can be regarded as an extension of the symmetrical Z-complementary code set (SZCCS). The available construction of SZCCS has a limitation on the set size, with a maximum set size of 8. The proposed generalized Boolean function based construction can generate CZCSS of length in the form of non-power-of-two with variable set size $2^{n+1}$, where each code has $2^{n+1}$ constituent sequences. The proposed construction also yields cross Z-complementary pairs and cross Z-complementary sets with a large number of constituent sequences compared to the existing work.

cs.IT↗

arXiv2022

Transitivity on subclasses of bipartite graphs

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge set, respectively. For two disjoint subsets $A$ and $B$, we say $A$ dominates $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$ where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive partition of a given graph with the maximum number of partitions. It was known that the decision version of \textsc{Maximum Transitivity Problem} is NP-complete for general graphs, which was proved by Hedetniemi et al. [Iterated colorings of graphs, \emph{Discrete Mathematics}, 278, 2004]. This paper first strengthens the NP-completeness result by showing that this problem remains NP-complete for perfect elimination bipartite graphs. On the other hand, we propose a linear-time algorithm for finding the transitivity of a given bipartite chain graph. We then characterize graphs with transitivity at least $t$ for any integer $t$. This result answers two open questions posed by J. T. Hedetniemi and S. T. Hedetniemi [The transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018].

cs.DM↗

arXiv2022

A Direct Construction of GCP and Binary CCC of Length Non Power of Two

Golay complementary pairs (GCPs) and complete complementary codes (CCCs) have found a wide range of practical applications in coding, signal processing and wireless communication due to their ideal correlation properties. In fact, binary CCCs have special advantages in spread spectrum communication due to their simple modulo-2 arithmetic operation, modulation and correlation simplicity, but they are limited in length. In this paper, we present a direct construction of GCPs, mutually orthogonal complementary sets (MOCSs) and binary CCCs of non-power of two lengths to widen their application in the recent field. First, a generalised Boolean function (GBF) based truncation technique has been used to construct GCPs of non-power of two lengths. Then Complementary sets (CSs) and MOCSs of lengths of the form $2^{m-1}+2^{m-3}$ ($m \geq 5$) and $2^{m-1}+2^{m-2}+2^{m-4}$ ($m \geq 6$) are generated by GBFs. Finally, binary CCCs with desired lengths are constructed using the union of MOCSs. The row and column sequence peak to mean envelope power ratio (PMEPR) has been investigated and compared with existing work. The column sequence PMEPR of resultant CCCs can be effectively upper bounded by $2$.

cs.IT↗

arXiv2020

Grid Obstacle Representation of Graphs

The grid obstacle representation, or alternately, $\ell_1$-obstacle representation of a graph $G=(V,E)$ is an injective function $f:V \rightarrow \mathbb{Z}^2$ and a set of point obstacles $\mathcal{O}$ on the grid points of $\mathbb{Z}^2$ (where no vertex of $V$ has been mapped) such that $uv$ is an edge in $G$ if and only if there exists a Manhattan path between $f(u)$ and $f(v)$ in $\mathbb{Z}^2$ avoiding the obstacles of $\mathcal{O}$ and points in $f(V)$. This work shows that planar graphs admit such a representation while there exist some non-planar graphs that do not admit such a representation. Moreover, we show that every graph admits a grid obstacle representation in $\mathbb{Z}^3$. We also show NP-hardness result for the point set embeddability of an $\ell_1$-obstacle representation.

cs.CG↗