# Dominating Number on Icosahedral-Hexagonal Network

Authors: Miroslava Mihajlov Carević
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we deal with the dominating set and the domination number on an icosahedral-hexagonal network. We will consider all cases of successive halving of the edges of triangles that are the sides of icosahedrons and thus obtain icosahedral-hexagonal networks.


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## Body

## 1. Introduction

The German weather service has developed an operational global numerical model for weather forecasting, called GME, based on the icosahedral-hexagonal network. In this paper [1], the authors describe this network and provide estimates of the accuracy of its application. In order to generate a network in the sphere, a regular icosahedron was constructed (Figure 1) so that 2 of the 12 verticals passing through the 12 vertices of the icosahedron coincide with the north and south poles.Figure 1
Regular icosahedron (the image is taken from paper [1]).By successive halving of the edges of the triangles that are the sides of the icosahedron, new triangles are formed (Figure2). The parameter n represents the number interval on the starting triangle of the icosahedral network.Figure 2
Hunting the edges of the icosahedron (the image is taken from paper [1]).The icosahedral network develops into a hexagonal network consisting of 10 rhombuses, each of which is composed of 2 triangles from the icosahedral network (Figure3).Figure 3
Icosahedral-hexagonal network GME (the image is taken from paper [1]).The icosahedral-hexagonal network was first introduced into meteorological modeling in 1968. In recent years, there has been a growing interest of researchers in this type of network.In the previous period, the icosahedral network was the subject of analysis by the author of this paper. In paper [2], the generating function for icosahedral figurative numbers is constructed.In paper [3], the results of our research on the dominating set and domination number of an icosahedral-hexagonal network without halving the edges of the triangles that are the sides of an icosahedron are presented.In this paper, we consider an icosahedral-hexagonal network with successive halving of the edges of the triangles and determine domination numbers for the network thus obtained.


## 2. Materials and Methods

Like usual in mathematics, we denote the vertex set and edge set of the graphG by V (G) and E (G), respectively.A subsetD of V (G) is called a t-dominating set, if for every vertex y not in D, there exists at least one vertex x in D, such that d (x, y) ≤ t. With d (x, y), we denote distance between the vertices x and y.The number of elements of the smallestt-dominating set is called the t-domination number and is denoted by γt. For t = 1, 1-dominating set is called dominating set and 1-domination number is also called domination number and denoted by γ.In the previous period,t-dominance was investigated on hexagonal [4–7], rectangular [8], and triangular [9] cactus chains. Research was also conducted for R-domination in graphs [10], k-dominating number of Cartesian products of two paths [11], paid domination in graphs [12], and total and double-total domination number on the hexagonal network [13].


## 3. Results and Discussion

In this section, we are going to consider 1-domination (or only domination) on rhomboidal networks composed of triangles that are sides of an icosahedron. We are going to analyze all cases of successive halving of the edges of triangles onn = 2k parts where k ∈ N.We denoted the rhomboidal-triangular network withm rows and c columns with Rm, c and with Rmc vertices in this network. The initial rhomboidal-triangular network consists of 2 rhombuses, i.e., 4 triangles from the icosahedral-hexagonal network (Figure 4). The complete icosahedral-hexagonal network consists of 5 rhomboidal-triangular networks.Figure 4
Rhomboidal-triangular networkR2,3.In paper [3], it was proved that the 1-domination number for R2, 3 is equal to 2. By halving the edges of the triangles in the rhomboidal-triangular network R2, 3, we obtain different cases that are the subject of the analysis in this paper. We denote the number of divisions of the triangles edges in the network R2, 3 by n.Lemma 1.
Forn = 2, the domination number is γ (R3,5) = 3.Proof.
By dividing the edges of the triangle in the networkR2,3 into 2 parts, we obtain the network R3,5 (Figure 5).
Let us prove that the minimum dominating set forR3,5 is the set D1 = {R21, R23, R25}.
NodeR21 dominates nodes R11, R12, R22, and R31. Node R23 dominates nodes R13, R14, R22, R24, R32, and R33. Node R25 dominates nodes R15, R24, R34, and R35. Hence, these 3 nodes dominate all nodes in the network R3, 5. Suppose that the set D1 is not a minimal dominating set but that there is a set D1′ of less cardinality that dominates the network R3,5. The set D1′ would then have 2 nodes that would dominate over the 13 remaining nodes in the network R3,5. One node Rij in the grid can dominate a maximum of 6 nodes of the hexagon in the center of which is the node Rij. Hence, two nodes can dominate a maximum of 12 nodes. These would be nodes R22 and R24. However, they cannot dominate over 13 nodes (they are not dominant for nodes R11 and R35). Hence, the set D1' = {R22, R24} is not dominant for the network R3,5. Similarly, it is proved that no other set of 2 nodes can dominate the entire network R3,5. We can conclude that D1 is the minimum dominating set for the network R3,5 and its cardinality is 3, therefore γ (R3,5) = 3.Figure 5
Rhomboidal-triangular networkR3,5.Lemma 2.
Forn = 22, the domination number is γ (R5,9) = 8.Proof.
By dividing the edges of the triangle in the networkR2,3 into 4 parts we obtain the network R5,9 (Figure 6).
Let us look at the setD2 = {R14, R21, R26, R28, R42, R44, R49, R56}. Nodes R42 and R44 dominate nodes R3i and R5j for i = 2, 3, 4, and 5 and j = 1, 2, 3, and 4. Node R21 dominates nodes R11, R12,R22, and R31. Node R14 dominates nodes R13, R15,R23, and R24. Similarly, nodes R26, R28,R56, and R49 dominate the nodes in the second rhombus including nodes R25 and R55 that were not covered by the previous dominance. So the set D2 dominates all nodes in the network R5,9.
Let us prove thatD2 is the minimal set that dominates over all nodes in the network R5,9. Suppose there is a set D2' whose cardinality is less than 8 (supposing 7) and which dominates the network R5,9. As mentioned earlier, one node in a network can dominate a maximum of 6 nodes that form a hexagon around it. Note that the network R5,9 is composed of 8 hexagons and 2 more nodes R11 and R59. The seven adjacent hexagons in the network have common nodes over which they dominate. The total number of nodes without repeating dominance for 7 hexagons in the grid is 31 (6 + 3 · 5 + 4 + 2 · 3). If we add to this number the number of nodes that dominate them (7), we get the sum of 38. As the network R5,9 has 45 nodes, there are still 7 uncovered nodes, which leads us to the conclusion that there is no set D2' whose cardinality is less than 8 and which dominates the network R5,9. Hence, D2 is the minimum dominating set for the network R5,9 and its cardinality is 8, therefore γ (R5,9) = 8.Figure 6
Rhomboidal-triangular networkR5,9.Lemma 3.
Forn = 23, the domination number is γ (R9,17) = 32.Proof.
By dividing the edges of the triangle in the networkR2,3 into 8 parts, we get the network R9,17 (Figure 7).
Observe that the networkR9,17 is composed of 4 networks R5,9. By applying the results expressed in Lemma 2, we form a set:(1)D3=R1i,i=4,12∪R2j,j=1,6,8,9,14,16∪R4k,k=2,4,9,10,12,17∪R5m,m=4,6,12,14∪R6n,n=1,6,8,9,14,16∪R8p,p=2,4,9,10,12,17∪R9q,q=6,14.
Based on the proof in Lemma2 and the fact that the network R9,17 is composed of 4 networks R5,9, we conclude that the set D3 dominates over all nodes in the network R9,17. Suppose that there is a set D3' of less cardinality that dominates the network R9,17. Let us take that D3′=D3/R66. Then, the distance of node R76 from the nearest nodes in the network is equal to 2. So there is a node in the network over which no node from the set D3' dominates. We come to the same conclusion by dropping any node from the set D3. Based on this, we conclude that D3 is the minimum dominating set for the R9,17 network. Since the cardinality of the set D3 is equal to 32, it follows that γ (R9,17) = 32.Figure 7
Rhomboidal-triangular networkR9,17.Lemma 4.
Forn=2kik∈N/1, the dominating number is γ (Rn+1, 2n+1) = 22k−1.Proof.
We derive the proof by using the principle of mathematical induction. Fork = 2, we obtain the statement expressed in Lemma 2 where it is also proved. Suppose that the statement holds for k = m, m ∈ N ^ m > 2, i.e., γ (Rn+1, 2n+1) = 22m−1 where n = 2m. Thus, we assumed it was γR2m+1,2m+1+1=22m−1.
Let us prove that the statement holds fork = m + 1 i.e., γ (Rn+1, 2n+1) = 22m+1 where n = 2m+1. We need to prove that γR2m+1+1,2m+2+1=22m+1.
Fork = m + 1, i.e., n = 2m+1, we obtain the division of the edges of the triangle in the initial network R2,3 on 2m+1 parts and network R2m+1+1,2m+2+1 which is composed of 4 networks R2m+1,2m+1+1.
As it was proved in Lemma3, it is not possible to remove any nodes from the dominating set for the network R2m+1+1,2m+2+1 which we obtain by uniting 4 dominating sets for the network R2m+1,2m+1+1 for m∈N ^ m > 2. It follows that it is(2)γR2m+1+1,2m+2+1=4⋅γR2m+1,2m+1+1=4⋅22m−1=22⋅22m−1=22m+1,
which needed to be proven.


## 4. Main Results

In this section, we deal with the dominance of the icosahedral-hexagonal network shown in Figure8. This network is composed of 5 rhomboidal-triangular networks that we called “prongs” and are numbered from 1 to 5 as shown in Figure 8(a).Figure 8
Icosahedral-hexagonal network with the prongs and indices.
(a)(b)We marked the nodes in the icosahedral-hexagonal network withRmcp where p denotes the prong, m denotes the rows, and c denotes the column in the rhomboidal-triangular network of the prong p. The center of the icosahedral-hexagonal network is the node denoted by R1 (Figure 8(b)).Notice that some nodes in the network match:(3)R121≡R212;R122≡R213;R123≡R214;R124≡R215iR125≡R211.As in the previous section, we are going to analyze all cases of successive halves of the edges of triangles of the initial network (Figure8) on n = 2k parts where k ∈ N. We denote the icosahedral-hexagonal network created by halving the edges of triangles into n parts by IHn+1,2n+1. In the next part, we deal with determining the domination number γ (IHn+1,2n+1) for the icosahedral-hexagonal network.Theorem 1.
Forn = 2, the domination number for the icosahedral-hexagonal network is as follows:(4)γIH3,5=11.Proof.
To prove this statement, we will use the results obtained in the previous section. By dividing the edges of the triangles in the icosahedral-hexagonal network shown in Figure8 into 2 equal parts, we obtain the icosahedral-hexagonal network shown in Figure 9.
The obtained icosahedral-hexagonal network is composed of 5 networksR3,5 located in its prongs. In Lemma 1, it is proved that the dominating number for the network R3,5 is equal to 3. Based on Lemma 1, the set D=R21p,R23p,R25p,p=1,2,3,4,5 is the dominating set on the icosahedral-hexagonal network for n = 2. But it is not the minimum dominating set. Node R1 dominates over 5 nodes in its environment, and thus, it is possible to remove nodes R211,R212,R213,R214, and R215. Hence, the minimum dominating set for this network is the set D′=R23p,R25p,p=1,2,3,4,5∪R1. Its cardinality is 11, so the dominance number for the icosahedral-hexagonal network is γ (IH3,5) = 11, which had to be proved.Figure 9
Icosahedral-hexagonal network IH3,5.Theorem 2.
Forn = 4, the dominating number for the icosahedral-hexagonal network is as follows:(5)γIH5,9=38.Proof.
By dividing the edges of the triangles in the icosahedral-hexagonal network IH2,3 into 4 equal parts, we obtain the icosahedral-hexagonal network IH5,9 shown in Figure 10.
The obtained icosahedral-hexagonal network is composed of 5 networksR5, 9 which are located in its prongs. In Lemma 2, it is proved that the domination number for the network R5, 9 is equal to 8.
Based on Lemma2, the set D=R14p,R21p,R26p,R28p,R42p,R44p,R49p,R56p,p=1,2,3,4,5 is the dominating set on the icosahedral-hexagonal network IH5, 9. But it is not a minimum dominating set. Node R211≡R125 can be removed because it dominates over nodes dominated by nodes R121, R411, and R215. Also, node R214≡R213 can be removed because it dominates over nodes dominated by R124 and R414iR213. No other node can be removed because some node in the network would be left without dominance. If we dropped the node R122≡R123, the node R223 would remain without dominance. The same is true for node R121≡R122. So among the 5 nodes that make up the first circle around the central node R1, it is possible to throw out only 2 nodes.
Let us observe other nodes in setD that fulfill the condition that their mutual distance is equal to 1. Such nodes have common nodes over which they dominate, so it is hypothetically possible to remove one of them. If we dropped the node R414≡R413, the nodes R233 and R243 would remain without dominance. If we dropped the node R424, the nodes R524 and R334 would remain without dominance. The same is true for the other nodes in the set D. Based on that, the dominating set for the network IH5,9 is the set D′=D/R211,R214.
Its cardinality is 5 · 8–2, i.e., 38. Thus,γ (IH5,9) = 38, which had to be proved.Figure 10
Icosahedral-hexagonal network IH5,9.Theorem 3.
Forn = 8, the domination number for the icosahedral-hexagonal network is as follows:(6)γIH9,17=158.Proof.
By dividing the edges of triangles in an icosahedral-hexagonal network IH2,3 by 8 equal parts, we obtain the icosahedral-hexagonal network IH9,17 shown in Figure 11.
The obtained icosahedral-hexagonal network is composed of 5 networksR9,17 which are located in its prongs. In Lemma 3, it was proved that the domination number γ (R9,17) = 32.
Based on Lemma3 set,(7)D3=R1ip,i=4,12∪R2jp,j=1,6,8,9,14,16∪R4kp,k=2,4,9,10,12,17∪R5mp,m=4,6,12,14∪R6np,n=1,6,8,9,14,16∪R8pp,q=2,4,9,10,12,17∪R9qs,s=6,14za,p=1,2,3,4,and5,which is the dominating set on the icosahedral-hexagonal network IH9, 17.
Similar to the previous, setD is not the minimum dominance set for network IH9, 17.
NodeR211≡R215 can be removed because it dominates over nodes dominated by nodes R121, R411, and R215. Also, node R214≡R213 can be removed because it dominates over nodes dominated by R124, R414, and R213.
Let us consider nodes in setD that satisfy the condition that their mutual distance is equal to 1 and the assumption that one of them is removed from set D. If we throw out node R422, node R332 would remain without dominance. If we dropped the node R415≡R414, the node R244 would remain without dominance. The same is true for the other nodes in the set D. Based on that, the dominating set for the network IH9, 17 is the set D′=D/R211,R214.
Its cardinality is 5 · 32–2, i.e., 158. Hence,γ (IH9, 17) = 158 which had to be proved.Figure 11
Icosahedral-hexagonal network IH9, 17.Theorem 4.
Forn=2kik∈N/1, the dominating number γ (IHn+1, 2n+1) = 5 · 22k−1 − 2.Proof.
Based on the results proved in Lemma4, the dominance number for the rhomboidal-triangular network γ (Rn+1, 2n+1) = 22k−1. The icosahedral-hexagonal network is composed of 5 rhomboidal-triangular networks that have a common start R1, where node R1 does not belong to the dominating set of the rhomboidal-triangular network. Based on that, we conclude that the dominating number of the icosahedral-hexagonal network is γ (IHn+1, 2n+1) ≤ 5 · γ (Rn+1, 2n+1). The dominance set on the icosahedral-hexagonal network IHn+1, 2n+1 is denoted by D. Since there are 5 nodes of the dominating set D at the distance 1 from the common beginning R1, we will consider the possibility of removing some of these 5 nodes from the set D. Analogous to the proof of Theorem 3, we conclude that, from set D it is possible to remove only 2 nodes among the 5 nodes that form the first circle around the central node R1. In addition to the common node R1, the rhomboidal-triangular networks in the icosahedral-hexagonal network touch parts of their edges. By removing any node from the tangent edges as well as any other node from the set D, at least one node in the network IHn+1,2n+1 would remain without dominance. Based on that, we conclude that it is γ (IHn+1,2n+1) = 5 · γ (Rn+1,2n+1) – 2. Thus, based on Lemma 4, we obtain γ (IHn+1,2n+1) = 5 · 22k−1 − 2 which needed to be proven.


## 5. Conclusion

Starting from the rhomboidal-triangular networkR2, 3 which is composed of 2 rhombuses, i.e., 4 triangles from the icosahedral-hexagonal network for which the dominance number is determined in [3], we determined the dominance number for the rhomboidal-triangular network Rm,c composed of m rows and c columns. The network Rm,c is formed by dividing the edges of the triangles in the initial network R2,3 into n = 2k parts where k ∈ N\{1}. We have proved that the dominating number is γ (Rn+1,2n+1) = 22k−1.By dividing the edges of the triangles in the icosahedral-hexagonal network IH2,3 by n = 2k equal parts, we obtained the icosahedral-hexagonal network IHn+1,2n+1. We determined the dominance number for the icosahedral-hexagonal network IH3,5 and then for IHn+1,2n+1 where n=2kandk∈N/1 as in the rhomboidal-triangular network. We proved that γ (IHn+1,2n+1) = 5 · 22k−1 − 2.In this paper, we determined the 1-dominating sets and the 1-domination numbers on an icosahedral-hexagonal network. In future works, we plan to explore and determinet-domination on the icosahedral-hexagonal grid for t > 1.



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