arXiv subjects

A. Senthil Thilak

Publications and source records attributed to A. Senthil Thilak.

6 recordsLinked to original sources

arXiv2026

A note on efficient k-limited broadcast domination in graphs

An efficient $k$-limited dominating broadcast, or $k$-ELDB, is a $k$-limited broadcast in which every vertex is dominated exactly once. This notion brings together efficient domination and limited broadcast domination in a common framework. For a graph $G$, we write $mcr(G)$ for the smallest integer $k$ for which $G$ admits a $k$-ELDB. For an admissible value $k\ge mcr(G)$, we denote by $γ_{ebk}(G)$ the minimum cost of a $k$-ELDB on $G$, and is called the $k$-efficient broadcast domination number of $G$. In this paper, we study these parameters from an algorithmic perspective with complexity analysis. We develop a dynamic programming algorithm for trees which, for fixed $k$, computes $γ_{ebk}(T)$ and thereby obtains a polynomial-time procedure for determining $mcr(T) $. In contrast, we prove that, for every fixed integer $k\ge 1$, deciding whether a graph admits a $k$-ELDB is NP-complete for arbitrary graphs. These results place efficient limited broadcast domination in a natural complexity framework, with trees forming a tractable class and arbitrary graphs remaining computationally hard.

cs.DS↗

arXiv2025

Efficient $k$-limited Dominating Broadcasts in Product Graphs

In a graph $ G $, a subset of vertices $ S $ is called an efficient dominating set (EDS) if every vertex in the graph is uniquely dominated by exactly one vertex in $ S $. A graph is said to be efficiently dominatable if it contains an EDS. Additionally, a function $ f: V(G) \rightarrow \{0, 1, 2, \dots, k\} $ is termed a $ k $-limited dominating broadcast if, for every vertex $ u \in V(G) $, there exists a vertex $ v $, with $ f(v) \geq 1$ such that $ d(u, v) \leq f(v) $. A vertex $u$ is said to be dominated by a vertex $v$. In this work, we unify these two concepts to explore the notion of efficient $k$-limited broadcast domination in graphs. A $ k $-limited dominating broadcast $f$ is called an efficient $k$-limited dominating broadcast ($k$-$ELDB$) if each vertex in the graph is dominated exactly once. The minimum value of $k$ for which the given graph $G$ has $k$-$ELDB$ is defined as $mcr(G)$. We prove determining $mcr(G)$ is NP-Complete for general graphs and explore the $mcr(G)$ values and other related parameters on standard graphs and their products.

math.CO↗

arXiv2024

Frame Scaling by Graphs

In this paper, we investigate the scalability of a given frame in $\mathbb{R}^n$ by using graphs. For each frame $ϕ$ in $\mathbb{R}^n$, we associate a simple undirected graph $G(ϕ)$ and use it to verify the scalability of $ϕ$. We provide some necessary conditions to test the scalability of a given frame. Finally, we study the scalability of some special classes of frames by using graphs.

math.FA↗

arXiv2023

Efficient Domination in Lattice graphs

Given a graph $G$, a subset $S$ of vertices of $G$ is an efficient dominating set ($EDS$) if $|N[v] \cap S|=1,$ for all $v\in V(G)$. A graph $G$ is efficiently dominatable if it possesses an $EDS$. The efficient domination number of G is denoted by F(G) and is defined to be $\max \left\{\sum_{v \in S}(1 + \operatorname{deg} v):\right.$ $\left.S \subseteq V(G)\right.$ and $\left.|N[x] \cap S| \leq 1, \forall~ x \in V(G)\right\}$. In general, not every graph is efficiently dominatable. Further, the class of efficiently dominatable graphs has not been completely characterized and the problem of determining whether or not a graph is efficiently dominatable is NP-Complete. Hence, interest is shown to study the efficient domination property for graphs under restricted conditions or special classes of graphs. In this paper, we study the notion of efficient domination in some Lattice graphs, namely, rectangular grid graphs ($P_m \Box P_n$), triangular grid graphs, and hexagonal grid graphs.

math.CO↗

arXiv2011

Design and Analysis of SD_DWCA - A Mobility based clustering of Homogeneous MANETs

This paper deals with the design and analysis of the distributed weighted clustering algorithm SD_DWCA proposed for homogeneous mobile ad hoc networks. It is a connectivity, mobility and energy based clustering algorithm which is suitable for scalable ad hoc networks. The algorithm uses a new graph parameter called strong degree defined based on the quality of neighbours of a node. The parameters are so chosen to ensure high connectivity, cluster stability and energy efficient communication among nodes of high dynamic nature. This paper also includes the experimental results of the algorithm implemented using the network simulator NS2. The experimental results show that the algorithm is suitable for high speed networks and generate stable clusters with less maintenance overhead.

cs.DM↗

arXiv2011

A Multi-Hop Weighted Clustering of Homogenous MANETs Using Combined Closeness Index

In this paper, a new multi-hop weighted clustering procedure is proposed for homogeneous Mobile Ad hoc networks. The algorithm generates double star embedded non-overlapping cluster structures, where each cluster is managed by a leader node and a substitute for the leader node (in case of failure of leader node). The weight of a node is a linear combination of six different graph theoretic parameters which deal with the communication capability of a node both in terms of quality and quantity, the relative closeness relationship between network nodes and the maximum and average distance traversed by a node for effective communication. This paper deals with the design and analysis of the algorithm and some of the graph theoretic/structural properties of the clusters obtained are also discussed.

cs.DM↗