A vertex \( v \in V(G) \) is said to be a self vertex switching of \( G \) if \( G \) is isomorphic to \( G^v \), where \( G^v \) is the graph obtained from \( G \) by deleting all edges of \( G \) incident to \( v \) and adding all edges incident to \( v \) which are not in \( G \). Two vertices \( u \) and \( v \) in \( G \) are said to be interchange similar if there exists an automorphism \( \alpha \) of \( G \) such that \( \alpha(u) = v \) and \( \alpha(v) = u \). In this paper, we give a characterization for a cut vertex in \( G \) to be a self vertex switching where \( G \) is a connected graph such that any two self vertex switchings, if they exist, are interchange similar.

Pavel Hrnciar and Alfonz Havier \([6]\) introduced a clever idea of transferring labeled pendant edges incident at a vertex of a graceful tree to some other suitable vertex of that tree, so that another graceful tree is obtained. This idea is further explored in this paper to generate graceful lobsters from a graceful caterpillar with \( n \) edges.

In this paper, we study the prime filters of a bounded pseudocomplemented semilattice. We extend some of the results of \([3]\) to pseudocomplemented semilattices. It is observed that the set of all prime filters \( \mathcal{P} \) of a pseudocomplemented semilattice \( S \) is a topology, and it is \( T_0 \) and compact. We also obtain some necessary and sufficient conditions for the subspace of maximal filters to be normal.

A node ranking problem is also called an \emph{ordered coloring problem} \([6]\), which labels a graph \( G = (V, E) \) with \( C: V \to \{1, 2, \ldots, k\} \) such that for every path between any two nodes \( u \) and \( v \), with \( C(u) = C(v) \), there is a node \( w \) on the path with \( C(w) > C(u) = C(v) \). The value \( C(v) \) is called the \emph{rank} or color of the node \( v \). Node ranking is the problem of finding the minimum \( k \) such that the maximum rank in \( G \) is \( k \). There are two versions of the node ranking problem: off-line and on-line. In the off-line version, all the vertices and edges are given in advance. In the on-line version, the vertices are given one by one in an arbitrary order (say \( v_1, v_2, \ldots, v_n \)) and only the edges of the induced subgraph \( \langle\{v_1, v_2, \ldots, v_i\}\rangle_G \) are known when the rank of \( v_i \) has to be chosen. This paper establishes the node ranking number of complete \( r \)-partite graphs for the off-line version and gives a tight bound for the on-line version with the algorithms to accomplish them in linear time.

Embeddings capabilities play a vital role in evaluating interconnection networks. Wirelength is an important measure of an embedding. As far as the most versatile architecture, the hypercube, is concerned, only approximate estimates of the wirelength of various embeddings are available. This paper presents an optimal embedding of the hypercube into a new architecture called \( k \)-cube necklace, which minimizes wirelength. In addition, this paper gives an exact formula for the minimum wirelength of the hypercube into \( k \)-cube necklace and thereby we solve completely the wirelength problem of the hypercube into \( k \)-cube necklace.

A labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \( (x, y) \) is the absolute value of the difference of the labels of \( x \) and \( y \). We say that a labeling of the vertices of a graph of order \( n \) is minimally \( k \)-equitable if the vertices are labeled with \( 1, 2, \ldots, n \) and in the induced labeling of its edges, every label either occurs exactly \( k \) times or does not occur at all. In this paper, we prove that Butterfly and Benes networks are minimally \( 2^r \)-equitable, where \( r \) is the dimension of the networks.

A \((2,2)\) packing on a graph \(G\) is a function \(f: V(G) \to \{0, 1, 2\}\) with \(f(N[v]) \leq 2\) for all \(v \in V(G)\). For a function \(f: V(G) \to \{0,1,2\}\), the Roman influence of \(f\), denoted by \(I_R(f)\), is defined to be \(I_R(f) = (|V_1|+|V_2|) + \sum_{v\in V_2} deg(v)\). The efficient Roman domination number of \(G\), denoted by \(F_R(G)\), is defined to be the maximum of \(I_R(f)\) such that \(f\) is a \((2,2)\)-packing. That is, \(F_R(G) = \text{max}\{I_R(f): f \text{ is a }(2,2)-\emph{packing}\}\). A \((2,2)\)-packing \(F_R(G)\) with \(F_R(G) = I_R(f)\) is called an \(F_R(G)\)-function. A graph \(G\) is said to be efficiently Roman dominatable if \(F_R(G) = n\), and when \(F_R(G) = n\), an \(F_R(G)\)-function is called an efficient Roman dominating function. In this paper, we focus our study on certain graphs which are efficiently Roman dominatable. We characterize the class of \(2 \times m\) and \(3 \times m\) grid graphs, trees, unicyclic graphs, and split graphs which are efficiently Roman dominatable.

The aim of this article is focused on developing an efficient algorithm for simulating Cellular Neural Network arrays (CNNs) using numerical integration techniques. The role of the simulator is that it is capable of performing raster simulation for any kind as well as any size of input image. It is a powerful tool for researchers to investigate the potential applications of CNN. This article proposes an efficient pseudo code for exploiting the latency properties of Cellular Neural Networks along with well known numerical integration algorithms. Simulation results and comparison have also been presented to show the efficiency of the numerical integration algorithms. It is observed that the Runge-Kutta (RK) sixth order algorithm outperforms well in comparison with the Explicit Euler, RK-Gill and RK-fifth order algorithms.

Image compression is the key technology in the development of various multimedia applications. Vector quantization is a universal and powerful technique to compress a data sequence, such as speech or image, resulting in some loss of information. In VQ, minimization of Mean Square Error (MSE) between code book vectors and training vectors is a non-linear problem. Traditional LBG types of algorithms used for designing the codebooks for Vector Quantizer converge to a local minimum, which depends on the initial code book. Memetic algorithms (MAs) are population-based meta-heuristic search approaches that have been receiving increasing attention in the recent years. These algorithms are inspired by models of natural systems that combine the evolutionary adaptation of a population with individual learning within the lifetimes of its members. It has shown to be successful and popular for solving optimization problems. In this paper, we present a new approach to vector quantization based on memetic algorithm. Simulations indicate that vector quantization based on memetic algorithm has better performance in designing the optimal codebook for Vector Quantizer than conventional LBG algorithm. The Peak Signal to Noise Ratio (PSNR) is used as an objective measure of reconstructed image quality.

Let \( G = (V, E) \) be a connected simple graph. Let \( u, v \in V(G) \). The detour distance, \( D(u, v) \), between \( u \) and \( v \) is the distance of a longest path from \( u \) to \( v \). E. Sampathkumar defined the detour graph of \( G \), denoted by \( D(G) \), as follows: \( D(G) \) is an edge-labelled complete graph on \( n \) vertices, where \( n = |V(G)| \), the edge label for \( uv \), \( u, v \in V(K_n) \), being \( D(u, v) \). Any edge-labelled complete graph need not be the detour graph of a graph. In this paper, we characterize detour graphs of a tree. We also characterize graphs for which the detour distance sequences are given.