Abstract
Wireless Sensor Network (WSN) is considered to be one of the fundamental technologies employed in the Internet of things (IoT); hence, enabling diverse applications for carrying out real-time observations. Robot navigation in such networks was the main motivation for the introduction of the concept of landmarks. A robot can identify its own location by sending signals to obtain the distances between itself and the landmarks. Considering networks to be a type of graph, this concept was redefined as metric dimension of a graph which is the minimum number of nodes needed to identify all the nodes of the graph. This idea was extended to the concept of edge metric dimension of a graph G, which is the minimum number of nodes needed in a graph to uniquely identify each edge of the network. Regular plane networks can be easily constructed by repeating regular polygons. This design is of extreme importance as it yields high overall performance; hence, it can be used in various networking and IoT domains. The honeycomb and the hexagonal networks are two such popular mesh-derived parallel networks. In this paper, it is proved that the minimum landmarks required for the honeycomb network HC(n), and the hexagonal network HX(n) are 3 and 6 respectively. The bounds for the landmarks required for the hex-derived network HDN1(n) are also proposed.
Highlights
IntroductionHoneycomb and Hexagonal networks [1] have been widely studied in various research domains, such as wireless sensor networks, wireless networks [2,3,4,5,6] and cellular networks, in order to study and analyze various issues like routing, [7,8,9] location management and target tracking [10,11,12], energy conservation [13,14,15], and interference estimation [16]
An edge metric generator (EMG) with the smallest size is referred to as an edge metric basis (EMB) for G, and its size is an edge metric dimension (EMD), which is denoted by edim(G)
In this paper the edge metric dimension of honeycomb and hexagonal networks were studied which could effectively be utilized by wireless sensor networks in a Internet of things (IoT) scenario, such as robot/drone navigation
Summary
Honeycomb and Hexagonal networks [1] have been widely studied in various research domains, such as wireless sensor networks, wireless networks [2,3,4,5,6] and cellular networks, in order to study and analyze various issues like routing, [7,8,9] location management and target tracking [10,11,12], energy conservation [13,14,15], and interference estimation [16]. A metric basis S of a connected graph G uniquely identifies all the nodes of G by means of distance vectors. For the metric basis {a, b} the edges ac and ad are equidistant from it while for the metric basic {c, d}, the edges ad and bd are equidistant and so on In this sense, a natural question is: Are there some sets of vertices/nodes which uniquely identify all the edges of a graph? An EMG with the smallest size is referred to as an edge metric basis (EMB) for G, and its size is an edge metric dimension (EMD), which is denoted by edim(G). Another useful approach for an EMG is as follows.
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