Domination Number and Secure Resolving Sets in Cyclic Networks

Abstract

Consider a robot that is navigating a graph-based space and is attempting to determine where it is right now. To determine how distant it is from each group of fixed landmarks, it can send a signal. We discuss the problem of determining the minimum number of landmarks necessary and their optimal placement to ensure that the robot can always locate itself. The number of landmarks is referred to as the graph's metric dimension, and the set of nodes on which they are distributed is known as the graph's metric basis. On the other hand, the metric dimension of a graph <I>G</I> is the minimum size of a set <i>w</i> of vertices that can identify each vertex pair of <I>G </I>by the shortest-path distance to a particular vertex in <i>w</i>. It is an NP-complete problem to determine the metric dimension for any network. The metric dimension is also used in a variety of applications, including geographic routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we investigate the exact value of the secure resolving set of some networks, such as trapezoid network, <I>Z</I>-(<i>P_n</i>) network, open ladder network, tortoise network and <img width="40" height="15" src="http://article.sciencepublishinggroup.com/journal/147/1471683/image001.png" /> network. We also determine the domination number of the networks, such as the twig network <i>T_m</i>, double fan network <i>F_2,n</i>, bistar network <i>B_n,n </i>and linear <i>kc</i>₄ – snake network.