Edge Metric Dimension of Honeycomb and Hexagonal Networks for IoT

: 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 HDN 1( n ) are also proposed.


Introduction
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]. These networks and application domains are applicable to the IoT as well [17][18][19][20][21][22][23][24]. The concept of metric basis and metric dimension for the purpose of robot navigation within a graph-structured framework was initially studied in [25]. Recently, the concept of robots and drones have also been proposed in such network structures for various application reasons [26]. For example, drones/robots might be used in wireless sensor networks to gather samples for maintenance inspection or regular data collection acting as a mobile sink. Similarly, they can also be used to recharge the batteries of dying sensors on-the-fly.
Suppose an IoT scenario where a robot/drone is dropped somewhere unknown in a network and can move from node to node. Some vertices in the graph behave as landmarks, and the robot can measure its distance in the graph to any landmark. The objective of this work is to find the fewest number of landmarks so that the robot can determine its current node by only using the distances to the landmarks. While dealing with navigation problems, networks are referred to as graphs G, the elements of the metric basis as landmarks, the minimum size of the metric basis as metric dimension dim(G) and the remaining vertices as nodes and identified the metric dimension of certain simple categories of graphs. The concept of metric basis also yielded significant applications in chemistry, drug design, image processing and combinatorial optimization [27].
But what if the robot takes edges into consideration while calculating distance to any landmark. Can the robot still measure its distance in the graph to any landmark such that the landmarks distinguish the edges instead of vertices? A metric basis S of a connected graph G uniquely identifies all the nodes of G by means of distance vectors. One could think that the edges of the graph are also identified by S with respect to the distances to S. However, this does not happen. For instance, Fig. 1 shows an example of a graph, where no metric basis uniquely recognizes all the edges of the graph. It can be observed that the graph G of Fig. 1 satisfies that dim(G) = 2. But for each one of the metric bases, there exist at least a pair of edges which is not distinguished by the corresponding metric basis as shown in Tab. 1. 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? To answer this, a new parameter was recently introduced in which the question is to whether there are some sets of nodes which can uniquely identify all the edges of a graph [28]. Let G = (V , E) be a connected graph. Let x ∈ V be a node and e = uv ∈ E be an edge, then the distance between the node x and the edge e is defined as dG (e, x) = min {dG (u, x), dG (v, x)}. A node x distinguishes any two edges e, f if dG (x, e) = dG (x, f). A set S ⊂ V is an edge metric generator (EMG) for G if for any two distinct edges e1 and e2, there exists at least one node v ∈ S such that the distances dG (e1, v) and dG (e2, v) are different. 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. Given an ordered set of nodes S = (s1, s2, . . . , sd ) of a connected graph G, for any edge e in G, the d-vector (ordered d-tuple) is referred to as r(e|S ) = (dG (e, s1), dG (e, s2), . . . , dG (e, sd )) as the edge metric representation of e with respect to S. In this sense, S is an EMG for G if and only if for every pair of different edges e1, e2 of G, r(e1|S ) = r(e2|S ) [29].
In [29], it was proved that the edim(G) = 1 if and only if it is a path. They also proved that paths Pn, cycles Cn and complete graphs Kn are families of graphs for whom dim(G) = edim(G).
The determination of EMD is NP hard problem. Hence, one has to consider particular classes of graphs. In [30,31], the author characterized the graphs for which edim(G) = n − 1. The EMD of some graph products was computed in terms of the graphs of the products [32].
In this paper the following have been proved: • The EMD of Honeycomb networks is 3.
• The EMD of Hexagonal networks is 6.
• The upper bound of two extensions of these networks called the Hex derived networks is proposed.
The rest of the manuscript is organized as follows. Section 2 provides the background of honeycomb and hexagonal networks along with their applications plus the related research in this domain. In Section 3, the edge metric dimension of the honeycomb and hexagonal networks are discussed where the resolving set for the HDN1(n) network is presented. The paper is concluded in the Section 4 while highlighting the limitations and future direction of the proposed work.

Background and Related Work
The honeycomb networks are built by recursively using the hexagonal tessellation [33]. The honeycomb network can be easily built from simple hexagons in various ways. HC(1) is just a simple hexagon. Adding one hexagon to each edge of this hexagon gives us HC (2). Similarly, HC(n) can be obtained in the same manner. The network cost, defined as the product of degree and diameter, is better for honeycomb networks than for the two other families based on square (mesh-connected computers and tori) and triangular (hexagonal meshes and tori) tessellations. Honeycomb networks provide simple and optimal routing, broadcasting, and semigroup computation algorithm. Honeycomb networks have major applications in mobile base stations, sensor networks, image processing, and computer graphics. The topological properties of honeycomb networks, their routing as well as torus honeycomb networks have been studied in [33].
On the other hand, the hexagonal network which was introduced in [34] is made of a minimum of six triangles and denoted by HX (2) whereas HX (1) does not exist. HX (3) comes from adding a series of triangles around HX (2). This pattern can be extended to HX (n). It is easy to see that n is the number of nodes on any one of the outermost edges of the hexagonal network, see  Multiprocessor interconnection networks are often required to connect thousands of homogeneously replicated processor-memory pairs, each of which is called a processing node. Instead of using a shared memory, all synchronization and communication between processing nodes for program execution is often done via message passing. Design and use of multiprocessor interconnection networks have recently drawn considerable attention due to the availability of inexpensive, powerful microprocessors and memory chips [34,35]. Regular plane tessellations can be easily constructed by repeating regular polygons. This design is of extreme importance for direct interconnection networks as it yields high overall performance.
A lot of research followed this and a host of generators, such as [36,37], were introduced. However, these generators primarily focused on the differentiation of the nodes. The idea was that a network in which a metric generator can distinguish the vertices behaves like an accurate surveillance activated network. But, if there is an intruder which accesses the network not through its vertices, but by using some connections between them (edges), then such intruder could not be identified, and in this sense, the surveillance fails in its commitment and some extra property is required in the network to be used effectively for this purpose [38]. The honeycomb and hexagonal networks along with the security are studied in [39]. Hence, such limitation can cause a major disadvantage from the application point of view. It is this shortcoming, that motivated the authors to explore a unique edge metric generator for this family of networks.

Result Discussions and Analysis
In this section, the EMD of the Honeycomb and hexagonal networks will be computed. The bounds for one of the hex derived networks will also be proposed. It is interesting to find that the EMD of honeycomb network and the hexagonal network are constant. For the hex-derived networks, it still needs to be seen whether an exact value for the EMD can be obtained or not.

Edge Metric Dimension of the Honeycomb Network
In this subsection, the EMD will be computed for the Honeycomb network HC(n), It is interesting to note that the EMD is the same as the metric dimension of the honeycomb network. This will provide a partial answer to the question about the families of graphs for which the EMD and the usual metric dimension are same. In order to prove the EMD for the honeycomb network, first the paths for this network are defined. Let αPi, βPj and γ Pk be paths defining HC(n) such that 0 ≤ i, j, k ≤ 2n − 1 as shown in the Fig. 3. It is easy to see that each edge is the intersection of exactly two different paths. Proof : This proof consists of two parts, where the first part is divided into three cases depending upon the paths, and the second part is divided into two cases. The first part of the proof is of forward inequality followed by proof of backward inequality thus leading to the equality itself. Since the honeycomb network is built recursively using the hexagonal tessellation, its properties exhibit symmetry across various dimensions. It is to explain these properties, that the proof is further divided into Cases, sometimes followed by sub-cases to further explain the unique cases underlying a particular pattern.
It will be proved that R = {α, β, γ } is a resolving set for HC(n). As one can see from Fig. 3, that HC(n) network consists of three types of edges with respect to each node in R. For simplicity, three types of edges with respect to α will be taken. The case of the edges with respect to the other nodes of R can be resolved in a similar manner. Let e1 and e2 be any two edges of HC(n). The following cases may ensue, depending on the types of edges.
Case 1: If both edges e1 and e2 belong to the same path αPi for some i, then they will be distinguished by α itself.
Case 2: If both edges e1 and e2 are parallel such that they lie between two consecutive paths αPi and αPi + 1 for some i, that is e1 and e2 does not completely lie on any of these two consecutive paths, then again, they will be distinguished by α.
Case 3: If e1 and e2 lie on two different paths αPi and αPj such that i = j, then all such pair of edges e1 and e2 will be automatically distinguished by α except the pair of edges for whom d (e1, α) = d(e2, α). In this situation, two possibilities may arise, i.e., : Subcase 3a: If e1, e2 ∈ βPk for some k, then we are in Case 1 for the node β. Subcase 3b: If e1, e2 / ∈ βPk for some k, then this happened only if e1 ∈ βP2k and e2 ∈ βPl + 1 for some k and l, then it is easy to see that d(e1, γ ) < d(e2, γ ) and both edges will be distinguished by γ . Hence, the edim(HC(n)) ≤ 3.
Conversely, it can be proved that the edim(HC(n)) ≥ 3. On the contrary, suppose that there is a resolving set of cardinality two, say S = {a, b} ∈ V (HC(n)). Since each node of HC(n) lies on some paths of α, β, γ , let us consider all the nodes lying on the paths indexed by γ . So, if there exits i and j such that a ∈ γ Pi and b ∈ γ Pj, then the following two cases are to be considered: Case 2: For i = j, let a ∈ γ Pi and b ∈ γ Pj such that they also lie on αPr or βPs. Then we will be in Case 1 for α or β. If a ∈ γ Pi and b ∈ γ Pj such that they do not lie on αPr or βPs, then the following possibilities may ensue: • Hence, the edim(HC(n)) > 3 and the proof is finished.

Edge Metric Dimension of the Hexagonal Network
In order to prove this, the concept of the co-ordinate system of the hexagonal network is presented as adapted by Nocetti et al. [39]. He introduced the three axes, X , Y and Z at an angle of 120 along the sides of any triangle within a hexagonal network as shown in the Fig. 4. In this section, the EMD for the hexagonal network HX (n) will be computed. It is interesting to note that the EMD is larger than the metric dimension of the hexagonal network. This will once again provide a partial answer to the question raised in [40] about the classification of families for which the EMD is larger than the usual metric dimension. The concept of edge neighborhood of a node and state some properties of the same are introduced now. For any node v ∈ V (HX (n)) and e ∈ E(HX (n)), the edge neighborhood of a node v is defined as For the convenience of describing the edges, the lines parallel to X -axis are referred to as X -lines and follow the same concept for defining Y -lines and Z-lines. Furthermore, a segment of any X , Y or Z line is referred to as S X , S Y and S Z respectively as shown in the Fig. 4. The central node is labeled by o and the six corner nodes of the outer most hexagon as {α, β, γ , θ, σ , ψ} as shown in the Fig. 5. Proof. We will prove that R = {α, β, γ , θ , σ , ψ} is a resolving set for HX (n).
The proof is presented in two steps. In the first step, it is proved that if e1 and e2 are two edges such that e1 and e2 belong to two different triangles, then they are distinguished by one of the nodes in R. In the second step, it is further proved that if e1 and e2 are two edges belonging to the same triangle, then once again they are distinguished by some node in R.
Case 1: Let e1 and e2 by two edges belongs to two different triangles. There are two subcases to consider depending upon the types of triangles.
Subcase 1a: Let e1 and e2 belong to two non-adjacent triangles. For simplicity let us take the two triangles αoψ and θ oγ such that e1 ∈ αoψ and e2 ∈ θ oγ . Then e1 ∈ E Np(α) such that 0 ≤ p ≤ n − 1 and e2 ∈ E Np(α) such that n − 1 ≤ p ≤ 2n − 2. Hence, at all times e1 and e2 will be distinguished by α ∈ R except when p = n − 1. In such a case, it is easy to see that e1 and e2 will be clearly distinguished by ψ.
Subcase 1b: Let e1 and e2 belong to two adjacent triangles. If e1 and e2 lie on the shared side of the two adjacent triangles, then they will fall under the restrictions of Case 2 discussed below. However, if the edges e1 and e2 are chosen such that e1 ∈ αoψ and e2 ∈ ψoσ such that they do not lie on the shared side then it is observed that e1 ∈ E Np(α) such that 0 ≤ p ≤ n − 1 and e2 ∈ E Np(α) such that n − 1 ≤ p ≤ 2n − 3. Hence, e1 and e2 will be distinguished by α ∈ R except when p = n − 1. In such a case, it is easy to see that e1 and e2 will be clearly distinguished by σ . Symmetry leads to similar reasoning for the cases of any pair of edges belonging to any pair of triangles in the network.
Case 2: Let the two edges e1 and e2 belong to the same triangle, say ψoσ . See Fig. 5. In this case, the following is observed: Subcase 2a: If the two edges e1 and e2 lie on the same segment of one of the co-ordinate axes, say S Z of the Z-axis. Then it is easy to see that the two edges e1 and e2 are distinguished by ψ and γ . Similarly, if the edges belonged to S X or S Y , then they will be distinguished by α and θ or β and σ respectively.
Subcase 2b: If two edges e1 and e2 are parallel to each other and lie on the segments of one of the axes, say S Xi and S Xj for i = j. Then it is clear to note that there exists distinct indices p and q such that e1 ∈ E Np(β) and e2 ∈ E Nq(β) and hence the edges are distinguished by β. Similar observations will be observed if the edges belonged to S Y or S Z . Subcase 2c: If the two edges e1 and e2 lie on segments of two different axes, then the following two possibilities arise: a) If two edges are incident on a node, for simplicity, it can be assumed that e1 = bd ∈ S Zi for some i, then the following two possibilities arise: • If e2 = ab ∈ S Xj for some j, then e1 and e2 will be distinguished by α.
• If e2 = dc ∈ S Xl for some l, then e1 and e2 will be distinguished by β.
Similarly, all other edges with a common node can be distinguished in a similar manner.
b) If e1 e2 are not incident edges, then they will be disguised by α except if e1, e2 ∈ E Nr (α). In such a case, if for simplicity let e1 ∈ S X , then e1, e2 will be distinguished by β because they will lie in different neighborhoods of β as d(e1, e2) ≥ 1. Hence, edim(HX (n)) ≤ 6.
Thus, all sets of 5 corner nodes of the HX (n) network generate similar equidistant edges. Hence, the edim(HX (n)) ≥ 6 which completes the proof.

Hex Derived Networks
In [41] two hex derived networks were introduced as they have much better connectivity as compared to the honeycomb and hexagonal networks. These are HDN1(n) and HDN2(n), for n ≥ 2 as shown in Fig. 6. The HDN1(n) is a planar graph and was shown to have 9n 2 − 15n + 7 vertices and 27n 2 − 51n + 24 edges with a diameter of 2n − 2. The HDN2(n) graph was shown to have 9n 2 − 15n + 7 vertices and 36n 2 −72n+36 edges with a diameter of 2n−2. These two architectures HDN1(n) and HDN2(n) have a few advantages over the hexagonal and honeycomb networks. The vertex-edge ratio of HDN1(n) and HDN2(n) are the same as that of hexagonal and honeycomb networks. However, both these hexagonal and honeycomb networks are simulated by these hex derived networks with no extra cost. This is possible since the hex derived networks contain hexagonal and honeycomb. HDN1(n) is planar, and it accommodates in a given space more processors and wires than hexagonal and honeycomb. In [42] it was proved that the metric dimension of HDN1(n) and HDN2(n) was between 3 and 4. In [43] it was proved that the metric dimension of HDN1(n) = 4. In this paper, a resolving set for the network of HDN1(n) was formulated which would make an upper bound of the minimum resolving set for HDN1(n). The resolving set R contains all the nodes of each Hexagon inside the network, except the central node. Hence, the EMD has to be greater than or equal to 12 and less than |R|, where |R|, depends on the types of n; even or odd.
Here |R|, represents the upper bound of the proposed resolving set for the hex derived network HDN1(n), and k is any positive integer.
The comparison of the research is summarized in Tab. 2. It is observed that the Metric dimension of HC(n) proposed in [41] and the edge metric dimension of these networks as proved in this research are the same. However, this research shows that the edge metric dimension of the HX(n) is greater than the metric dimension of HX(n) which was found out to be 3 in [41]. While [42] was able to give a bound for both the hex derived networks and [43] found the exact value for HDN1(n) networks and this research is able to find the bounds for the HDN1(n) networks.  [3,4] 12 ≤ edim ≤ |R|

Conclusion and Future Work
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 IoT scenario, such as robot/drone navigation. After representing the network via a graph in terms of robots and landmarks, a robot could measure its distance in the graph to any landmark. The objective was to enable robot in finding out the fewest number of landmarks in order to determine its current node thereby using only the distances to the landmarks. It was proved that the minimum landmarks required for a honeycomb network HC(n) and a hexagonal network HX (n) are 3 and 6, respectively. The bounds were proposed for the landmarks required for the hex-derived network HDN1(n), i.e., by finding out the resolving set for the network of HDN1(n). However, the exact values of the number of landmarks required for hex derived networks are still unfound and the methodology proposed for the number of landmarks required for hex networks cannot be naturally extended to hex derived networks. Since the hex derived networks are more efficient in an IoT scenario, exact values for the number of landmarks for these hex-derived networks point towards a future direction of research.