An efficient representation of Benes networks and its applications

Abstract

The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly-like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V , E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.

Computing topological indices of Sudoku graphs

“…Imran et al studied various degree based topological indices for various networks like silicates, hexagonal, honeycomb and oxide in [14]. For further study of topological indices of various graph families see, [2,9,10,12,13,15,18,[22][23][24][25]29,31,37]. …”

Resolving-power dominating sets

“…The basic idea of this proof is due to Khuller et al [7] and Manuel et al [8]. Question: Given a graph GðV; EÞ and an integer k, does there exist a set C of k vertices such that G admits a resolvingpower dominating set?…”

On Certain Connected Resolving Parameters of Hypercube Networks

“…It has been proved that the metric dimension problem is NP-hard [2] for general graphs. Manuel et al [9] have shown that the problem remains NP-complete for bipartite graphs. There are many applications of resolving sets to problems of network discovery and verification [1], pattern recognition, image processing and robot navigation [2], geometrical routing protocols [10], connected joins in graphs [11] and coin weighing problems [12].…”

“…There are many applications of resolving sets to problems of network discovery and verification [1], pattern recognition, image processing and robot navigation [2], geometrical routing protocols [10], connected joins in graphs [11] and coin weighing problems [12]. This problem has been studied for trees, multi-dimensional grids [2], Petersen graphs [13], torus networks [14], Benes networks [9], honeycomb networks [15], enhanced hypercubes [16] and Illiac networks [17].…”