On the Metric Dimension of Some Operation Graphs

CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 5(3) (2018), Pages 88-94 p-ISSN: 2086-0382; e-ISSN: 2477-3344

On the Metric Dimension of Some Operation Graphs Marsidi1, Ika Hesti Agustin2, Dafik3, Ridho Alfarisi4, Hendrik Siswono5 1Mathematics

Edu. Depart. IKIP PGRI Jember Indonesia Depart. University of Jember Indonesia 3Mathematics Edu. Depart. University of Jember Indonesia 4Elementary School Teacher Edu. Depart. University of Jember Indonesia 5Majoring in Early Childhood Edu. Depart. IKIP PGRI Jember Indonesia 2Mathematics

Email: [email protected], [email protected], [email protected] ABSTRACT Let 𝐺 be a simple, finite, and connected graph. An ordered set of vertices of a nontrivial connected graph 𝐺 is 𝑊 = {𝑤1 , 𝑤2 , 𝑤3 , … , 𝑤𝑘 } and the 𝑘-vector 𝑟(𝑣|𝑊) = (𝑑(𝑣, 𝑤1 ), 𝑑(𝑣, 𝑤2 ), … , 𝑑(𝑣, 𝑤𝑘 )) represent vertex 𝑣 that respect to 𝑊, where 𝑣 ∈ 𝐺 and 𝑑(𝑣, 𝑤𝑖 ) is the distance between vertex 𝑣 and 𝑤𝑖 for 1 ≤ 𝑖 ≤ 𝑘. The set 𝑊 called a resolving set for 𝐺 if different vertex of 𝐺 have different representations that respect to 𝑊. The minimum cardinality of resolving set of 𝐺 is the metric dimension of 𝐺, denoted by dim⁡(𝐺). In this paper, we give the local metric dimension of some operation graphs such as joint graph 𝑃𝑛 + 𝐶𝑚 , amalgamation of parachute, amalgamation of fan, and 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚). Keywords: metric dimension, resolving set, operation graphs.

INTRODUCTION All graphs in this paper are simple, finite and connected, for basic definition of graph we can see in Chartrand [1]. Chartrand [2] define the length of a shortest path between two vertices 𝑢 and 𝑣 is the distance 𝑑(𝑢, 𝑣) between two vertices in a connected graph G. An ordered set of vertices of a nontrivial connected graph 𝐺 is 𝑊 = {𝑤1 , 𝑤2 , 𝑤3 , … , 𝑤𝑘 } and the 𝑘-vector 𝑟(𝑣|𝑊) = (𝑑(𝑣, 𝑤1 ), 𝑑(𝑣, 𝑤2 ), … , 𝑑(𝑣, 𝑤𝑘 )) represent vertex 𝑣 that respect to 𝑊. The set 𝑊 called a resolving set for 𝐺 if different vertex of 𝐺 have different representations that respect to 𝑊. The minimum of cardinality of resolving set of G is the metric dimension of 𝐺, denoted by dim(𝐺) [3]. There are many articles explained about metric dimension such as [2], [4], [5], [6], and [7]. [8] defined a shackle graphs 𝑠ℎ𝑎𝑐𝑘(𝐺1 , 𝐺2 , … , 𝐺𝑘 ) constructed by nontrivial connected graphs 𝐺1 , 𝐺2 , … , 𝐺𝑘 such that 𝐺𝑖 and 𝐺𝑗 have no a common vertex for every 𝑖, 𝑗 ∈ [1, 𝑘] with |𝑖 − 𝑗| ≥ 2, and for every 𝑙 ∈ [1, 𝑘 − 1], 𝐺𝑙 and 𝐺𝑙+1 share exactly one common vertex (called linkage vertex) and the 𝑘 − 1 linking vertices are all different. [9] defined an amalgamation of graphs constructed from isomorphic connected graphs 𝐻 and the choice of the vertex 𝑣𝑗 as a terminal is irrelevant. For any 𝑘 positive integer, we denote such an amalgamation by 𝑎𝑚𝑎𝑙(𝐻, 𝑘), where 𝑘 denotes the number of copies of 𝐻. Proposition 1. [2] Let 𝐺 be a connected graph or order 𝑛⁡ ≥ ⁡2, then the following hold: a. 𝑑𝑖𝑚(𝐺) = 1⁡if and only if graph 𝐺 is a path graph b. 𝑑𝑖𝑚(𝐺) = 𝑛 − 1 if and only if graph 𝐺 is a complete graph c. For 𝑛 ≥ 3, 𝑑𝑖𝑚(𝐶𝑛 ) = 2 Submitted: 23 July 2018 Reviewed: 08 October 2018 DOI: http://dx.doi.org/10.18860/ca.v5i3.5331

Accepted: 30 November 2018

On the Metric Dimension of Some Operation Graphs

d. For 𝑛 ≥ 4, 𝑑𝑖𝑚(𝐺) = 𝑛 − 2 if and only if 𝐺 = 𝐾𝑝,𝑞 ⁡(𝑝, 𝑞 ≥ 1), 𝐺 = 𝐾𝑝 + ̅𝐾̅̅𝑞̅(𝑝 ≥ 1, 𝑞 ≥ 2). RESULTS AND DISCUSSION Theorem 2.1. For 𝑛 ≥ 2 and 𝑚 ≥ 7, the metric dimension of joint graph 𝑃𝑛 + 𝐶𝑚 is 𝑛

𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚 ) = ⌈ 2⌉ ⁡ + ⁡ ⌊

𝑚−1 2

⌋.

Proof. The joint of path and cycle graph, denoted by 𝑃𝑛 + 𝐶𝑚 is a connected graph with vertex set 𝑉(𝑃𝑛 + 𝐶𝑚 ) = {𝑥𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑛} ∪ {𝑦𝑙 ; ⁡1 ≤ 𝑙 ≤ 𝑚} and edge set 𝐸(𝑃𝑛 + 𝐶𝑚 ) = {𝑥𝑗 𝑦𝑙 ; ⁡1 ≤ 𝑗 ≤ 𝑛; ⁡1 ≤ 𝑙 ≤ 𝑚} ∪ {𝑥𝑗 𝑥𝑗+1 ; ⁡1 ≤ 𝑗 ≤ 𝑛 − 1} ∪ {𝑦𝑙 𝑦𝑙+1 ; ⁡1 ≤ 𝑙 ≤ 𝑚 − 1} ∪ {𝑦𝑛 𝑦1 }. The cardinality of vertex set and edge set, respectively are |𝑉(𝑃𝑛 + 𝐶𝑚 )| ⁡ = ⁡𝑛⁡ + ⁡𝑚 and |𝐸(𝑃𝑛 + 𝐶𝑚 )| = ⁡𝑛(𝑚 + 1) + 𝑚. 𝑛 𝑚−1 If we show that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚 ) = ⌈ 2⌉ ⁡ + ⁡ ⌊ 2 ⌋ for 𝑛⁡ ≥ 2 dan 𝑚 ≥ 7, then we will 𝑛

show the lower bound namely 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚 ) ≥ ⌈ 2⌉ ⁡ + ⁡ ⌊ 𝑛

𝑚−1

𝑚−1

2

⌋ − 1. Assume that 𝑑𝑖𝑚(𝑃𝑛 +

𝐶𝑚 ) < ⌈ 2⌉ ⁡ + ⁡ ⌊ 2 ⌋. This can be shown with take resolving set 𝑊 = {𝑥1 , 𝑦1 , 𝑦5 } so that it obtained the representation of the vertices 𝑥, 𝑦 ∈ 𝑉(𝑃2 + 𝐶7 ) respect to 𝑊. It can be seen that there is at least two vertices in 𝑃𝑛 + 𝐶𝑚 which have the same representation respect to 𝑊, one of them is 𝑟(𝑦4 |𝑊) ⁡ = (1, 2, 1) and 𝑟(𝑦6 |𝑊) ⁡ = (1, 2, 1) 𝑛 𝑚−1 such that we have the cardinality of resolving set of 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚 ) ≥ ⌈ 2⌉ ⁡ + ⁡ ⌊ 2 ⌋. 𝑛

Furthermore, we will prove that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚 ) ≤ ⌈ ⌉ ⁡ + ⁡ ⌊

𝑚−1

2

𝑛

resolving set 𝑊 = {𝑥𝑗 ; ⁡1 ≤ 𝑗 ≤ 2 ⌊ 2⌋ ; ⁡𝑖 ∈ 𝑜𝑑𝑑} ∪ {𝑦𝑙 ; ⁡1 ≤ 𝑙 ≤ 2 ⌊

⌋ with determine the

2 𝑚−1

⌉ 2⌈𝑛 2

2

⌋ ; ⁡𝑗 ∈ 𝑜𝑑𝑑}. So, we

2⌈𝑚−1⌉

𝑛

have the cardinality of resolving set of 𝑃𝑛 + 𝐶𝑚 namely |𝑊| = 2 ⁡ + 22 = ⌊ 2⌋ + ⌊ The representation of the vertices 𝑦 ∈ 𝐹𝑛 and 𝑥 ∈ 𝐹𝑛 respect to 𝑊 as follows. 𝑛 𝑚−1 𝑟(𝑥𝑗 |𝑊) ⁡ = ⁡ {(𝑎𝑖𝑗 ); ⁡1 ≤ 𝑗 ≤ 𝑛, 1 ≤ 𝑖 ≤ (⌈ 2⌉ + 1) + (⌊ 2 ⌋)}, where 0; 𝑓𝑜𝑟⁡𝑖 =

𝑗+1 ,1 2

2

⌋.

≤ 𝑗 ≤ 𝑛, 𝑗 ∈ 𝑜𝑑𝑑⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

𝑛

𝑎𝑖𝑗 =

𝑚−1

𝑛

1; 𝑓𝑜𝑟⁡ (⌈ 2⌉ + 1) ≤ 𝑖 ≤ (⌈ 2⌉ + 1) + (⌊ 𝑗

𝑚−1 ⌋) , 1 2

≤ 𝑗 ≤ 𝑛⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

𝑗

𝑜𝑟⁡𝑖 = 2, 2 ≤ 𝑗 ≤ 𝑛, 𝑗 ∈ 𝑒𝑣𝑒𝑛⁡𝑜𝑟⁡𝑖 = 2 + 1, 2 ≤ 𝑗 ≤ 𝑛, 𝑗 ∈ 𝑒𝑣𝑒𝑛

{2; 𝑓𝑜𝑟⁡𝑖, 𝑗 = 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑟(𝑦𝑙 |𝑊) = {(𝑎𝑖𝑗 ); ⁡1 ≤ 𝑗 ≤ 𝑚, 1 ≤ 𝑖 ≤ (⌈𝑛2⌉ + 1) + (⌊𝑚−1 ⌋)}, where 2 𝑛

0; 𝑓𝑜𝑟⁡𝑖 = (⌈ 2⌉) +

𝑗+1 ,1 2

≤ 𝑗 ≤ 𝑚 − 2, 𝑗 ∈ 𝑜𝑑𝑑

𝑛

1; 𝑓𝑜𝑟⁡1 ≤ 𝑖 ≤ (⌈ 2⌉) , 1 ≤ 𝑗 ≤ 𝑚 − 2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑎𝑖𝑗 = 𝑜𝑟⁡𝑖 = (⌈𝑛⌉) + 𝑗 , 2 ≤ 𝑗 ≤ 𝑚, 𝑗 ∈ 𝑒𝑣𝑒𝑛⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 2 2 𝑛

𝑜𝑟⁡𝑖 = (⌈ 2⌉) +

𝑗+1 2

+ 1, 2 ≤ 𝑗 ≤ 𝑚, 𝑗 ∈ 𝑒𝑣𝑒𝑛⁡⁡⁡⁡

{ 2; 𝑓𝑜𝑟⁡𝑖, 𝑗 = 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ It can be seen that every vertex in 𝑃𝑛 + 𝐶𝑚 have distinct representation respect to 𝑛 𝑚−1 𝑛 𝑊, such that the cardinality of resolvng set in 𝑃𝑛 + 𝐶𝑚 is ⌈ 2⌉ ⁡ + ⁡ ⌊ 2 ⌋ or 𝑑𝑖𝑚(𝐹𝑛 ) ≤ ⌈ 2⌉ ⁡ + ⁡⌊

𝑚−1 2

𝑛

⌋. Thus, we conclude that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚 ) ⁡ = ⁡ ⌈ 2⌉ ⁡ + ⁡ ⌊

Marsidi

𝑚−1 2

⌋ for 𝑛 ≥ 2 and 𝑚 ≥ 7.

∎

89

On the Metric Dimension of Some Operation Graphs

Fig 1. The Metric Dimension of Joint Graph 𝑃6 + 𝐶4 . Theorem 2.2. For 𝑛 ≥ 7, the metric dimension of amalgamation of parachute 6𝑚 𝑎𝑚𝑎𝑙⁡(𝑃𝐶7 , 𝑣, 𝑚) is 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣 = 𝐴, 𝑚)) = 2 . Proof. The amalgamation of parasut graph, denoted by 𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚) is a connected 𝑗 𝑗 graph with vertex set 𝑉(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) = {𝑥𝑖 ; ⁡1 ≤ 𝑖 ≤ 7; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑦𝑖 ; ⁡1 ≤ 𝑖 ≤ 𝑗 7; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝐴} and edge set 𝐸(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) = {𝐴⁡𝑥𝑖 ; ⁡1 ≤ 𝑖 ≤ 7; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 {⁡𝑥𝑖 ⁡𝑥𝑖+1 ; ⁡1 ≤ 𝑖 ≤ 6; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑦𝑖 ⁡𝑦𝑖+1 ; ⁡1 ≤ 𝑖 ≤ 6; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑥1 ⁡𝑦1 ; ⁡1 ≤ 𝑗 ≤ 𝑗 𝑗 𝑚} ∪ {⁡𝑥7 ⁡𝑦7 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. The cardinality of vertex set and edge set, respectively are |𝑉(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚))| ⁡ = 14m⁡ + ⁡1 and |𝐸(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚))| ⁡ = ⁡21𝑚. If we show that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) ≥ lower

bound

namely

6𝑚 2

r 𝑛⁡ = ⁡7, then we will show the best

𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) ≥

6𝑚

7𝑚 2

− 1.

Assume

that

𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) < 2 .⁡This can be shown with take resolving set 𝑊 = {𝑥11 , 𝑥41 , 𝑥61 , 𝑥12 , 𝑥42 , 𝑥62 , 𝑥13 , 𝑥43 , 𝑥63 , 𝑥14 , 𝑥44 , 𝑥64 } so that it obtained the representation of the vertices 𝑥, 𝑦 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) respect to 𝑊. It can be seen that there is at least two vertices in 𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 4) which have the same representation respect to 𝑊, one of them is 𝑟(𝑥31 |𝑊) ⁡ = (2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2) and 𝑟(𝑥51 |𝑊) ⁡ = (2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2) such 6𝑚 that we have the cardinality of resolving set of (𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) ≥ ⌈ 2 ⌉. Furthermore, we will prove that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) ≤ ⌈ 𝑗

𝑗

6𝑚 2

⌉ with determine the

resolving set 𝑊 = {𝑥𝑖 ; ⁡4 ≤ 𝑖 ≤ 7; ⁡2 ≤ 𝑗 ≤ 𝑚; ⁡𝑖 = 𝑜𝑑𝑑} ∪ {𝑥1 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. So, we have 𝑗 the cardinality of resolving set of 𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚) namely |𝑊| ⁡ = ⁡ |{𝑥𝑖 ; ⁡4 ≤ 𝑖 ≤ 7; ⁡2 ≤ 4

𝑗

6𝑚

𝑗 ≤ 𝑚; 𝑖 = 𝑜𝑑𝑑} ∪ {𝑥1 ; 1 ≤ 𝑗 ≤ 𝑚}| = (2 𝑚) ⁡ + 𝑚 = (

2

). The representation of the

vertices 𝑦 ∈ (𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚 = 4)) and 𝑥 ∈ (𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚 = 4)) respect to 𝑊 as follows. 6𝑚 𝑗 𝑗 𝑟(𝑥𝑖 |𝑊) = {(𝑎𝑖𝑘 ); ⁡1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚, 1 ≤ 𝑘 ≤ 2 }, where 𝑛

0; 𝑓𝑜𝑟⁡𝑘 = 1, 𝑘 = 2𝑖, 2 ≤ 𝑖 ≤ (⌊ 2⌋) , 1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑎𝑖𝑘 =

Marsidi

1; 𝑓𝑜𝑟⁡𝑘 =

𝑖+1 ,3 2

≤ 𝑖 ≤ 𝑛, 𝑖 ∈ 𝑜𝑑𝑑⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑖−1

1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 2 , 5 ≤ 𝑖 ≤ 7, 𝑖 ∈ 𝑜𝑑𝑑, 1 ≤ 𝑗 ≤ 𝑚 {2; 𝑓𝑜𝑟⁡1 ≤ 𝑗 ≤ 𝑚, 𝑘, 𝑖 = 𝑜𝑡ℎ𝑒𝑟⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

90

On the Metric Dimension of Some Operation Graphs

6𝑚+2

𝑟(𝑦|𝑊) = {(𝑎𝑖𝑘 ); ⁡1 ≤ 𝑖 ≤ 7, 1 ≤ 𝑘 ≤ 2 }, where 1; 𝑓𝑜𝑟⁡𝑘 = 3𝑗 − 2, 𝑖 = 1, 1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 6𝑚+2

2; 𝑓𝑜𝑟⁡𝑘 = 2 , 𝑖 = 1, 7, 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗 − 2, 𝑖 = 2,⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗, 𝑖 = 7, 1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 6𝑚+2

3; ⁡𝑓𝑜𝑟⁡𝑘 = 2 , 𝑖 = 2,6, 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗 − 2, 𝑖 = 3,⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗, 𝑖 = 6, 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑖 = 1,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑘 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑘 = 𝑎𝑖𝑘 =

6𝑚+2

6𝑚+2 ⁡𝑜𝑟⁡𝑖 2

= 7, 𝑘 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑘 =

6𝑚+2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡ 2

4; ⁡𝑓𝑜𝑟⁡𝑘 = 2 , 𝑖 = 3, 5, 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗 − 2, 𝑖 = 4,⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗, 𝑖 = 5, 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑖 = 2,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑘 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑘 = 5; ⁡𝑓𝑜𝑟⁡𝑘 =

6𝑚+2 ,𝑖 2

𝑜𝑟⁡𝑘 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑘 ≠

6𝑚+2 ⁡𝑜𝑟⁡𝑖 2

= 3, 3𝑗⁡𝑜𝑟⁡𝑘 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑘 ≠ 6𝑚+2 ,𝑖 2

6𝑚+2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡ 2 6𝑚+2 , 𝑖 = 3, 2

= 6, 𝑘 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑘 =

= 5⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

{6; 𝑓𝑜𝑟⁡𝑖 = 4⁡𝑎𝑛𝑑⁡𝑖 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑖 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑖 ≠

6𝑚+2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 2

It can be seen that every vertex in 𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚) have distinct representation 6𝑚 respect to 𝑊, such that the cardinality of resolvng set in 𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚) is 2 or 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) ≤

6𝑚 2

. Thus, we conclude that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7 , 𝑣, 𝑚)) ⁡ = ⁡

6𝑚 2

.

∎

Fig 2. The Metric Dimension of Amalgamation of Parachute 𝐴𝑚𝑎𝑙⁡(𝑃𝐶7 , 𝑣, 3). Theorem 2.3. For 𝑛 ≥ 6, the metric dimension of amalgamation of fan graph 𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚) is:

Marsidi

91

On the Metric Dimension of Some Operation Graphs

𝑛𝑚 − 1, 𝑓𝑜𝑟⁡𝑛⁡is⁡even 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝐴, 𝑚)) = {𝑛𝑚2 − 𝑚 , 𝑓𝑜𝑟⁡𝑛⁡is⁡odd⁡⁡⁡⁡ 2 Proof. The amalgamation of fan graph, denoted by 𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚) is a connected 𝑗 graph with vertex set 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) = {𝑥𝑖 ; ⁡1 ≤ 𝑖 ≤ 𝑛 − 1; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ 𝑗 𝑗 {𝑦𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑥𝑛𝑚 } and edge set 𝐸(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) = {⁡𝑥𝑖 ⁡𝑥𝑖+1 ; ⁡1 ≤ 𝑖 ≤ 𝑛 − 𝑗 𝑗 𝑗+1 2; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑦𝑗 𝑥𝑖 ; ⁡1 ≤ 𝑖 ≤ 𝑛 − 1; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑥𝑛−1 ⁡𝑥1 ; ⁡1 ≤ 𝑗 ≤ 𝑚 − 1} ∪ 𝑗+1 𝑚 {⁡𝑥𝑛−1 ⁡𝑥𝑛𝑚 } ∪ {𝑦𝑗 𝑥1 ; ⁡1 ≤ 𝑗 ≤ 𝑚 − 1} ∪ {𝑦𝑚 𝑥𝑛𝑚 }. The cardinality of vertex set and edge set, respectively are |𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚))| ⁡ = nm⁡ + ⁡1 and |𝐸(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚))| = 𝑚(2𝑛 − 1). 𝑛𝑚

If we show that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) =

2

− 1 for𝑛 ≥ 7 and 𝑛 is even, then we

will show the best lower bound namely 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) ≥

𝑛𝑚 2

𝑛𝑚

− 1. Assume that

𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) < 2 − 1.⁡This can be shown with take resolving set 𝑊 = {𝑥11 , 𝑥41 , 𝑥12 , 𝑥42 , 𝑥62 , 𝑥13 , 𝑥43 , 𝑥63 , 𝑥14 , 𝑥44 } so that it obtained the representation of the vertices 𝑗 𝑦 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐹6 , 𝑣 = 𝑦, 4)) and 𝑥𝑖 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 6, 4)) respect to 𝑊. It can be seen that there is at least two vertices in 𝑎𝑚𝑎𝑙(𝐹6 , 𝑣 = 𝑦, 4) which have the same representation respect to 𝑊, one of them is 𝑟(𝑥61 |𝑊) ⁡ = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2) and 𝑟(𝑥64 |𝑊) ⁡ = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2) such that we have the cardinality of resolving set of 𝑛𝑚 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) ≥ − 1. 2

Furthermore, we will prove that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚)) ≤

𝑛𝑚

2 𝑚 {𝑥𝑛 }

𝑗 {𝑥𝑖 ; ⁡4

− 1 with 𝑗

determine the resolving set 𝑊 = ≤ 𝑖 ≤ 𝑛; ⁡2 ≤ 𝑗 ≤ 𝑚; ⁡𝑖 = 𝑜𝑑𝑑} − ∪ {𝑥1 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. So, we have the cardinality of resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣 = 𝑦, 𝑚) namely |𝑊| = 𝑗

𝑛−2

𝑗

⁡|{𝑥𝑖 ; ⁡4 ≤ 𝑖 ≤ 𝑛; ⁡1 ≤ 𝑗 ≤ 𝑚; 𝑖⁡is⁡even} − {𝑥𝑛𝑚 } ∪ {𝑥1 ; 1 ≤ 𝑗 ≤ 𝑚}| = ( 𝑛𝑚

(

2

2

) 𝑚⁡ + 𝑚 − 1 =

− 1). The representation of the vertices 𝑦 ∈ 𝐹𝑛 and 𝑥 ∈ 𝐹𝑛 respect to 𝑊′ as follows. 𝑗

𝑗

𝑟(𝑥𝑖 |𝑊) = {(𝑎𝑖𝑘 ); ⁡1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚, 1 ≤ 𝑘 ≤ 𝑛

𝑛𝑚 2

− 1}, where

0; 𝑓𝑜𝑟⁡𝑘 = 1, 𝑘 = 2𝑖, 2 ≤ 𝑖 ≤ (⌊ 2⌋) , 1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑎𝑖𝑘 =

1; 𝑓𝑜𝑟⁡𝑘 = 𝑘=

𝑖−1 ,5 2

𝑖+1 ,3 2

≤ 𝑖 ≤ 𝑛, 𝑖 ∈ 𝑜𝑑𝑑, 1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

≤ 𝑖 ≤ 𝑛, 𝑖 ∈ 𝑜𝑑𝑑, 1 ≤ 𝑗 ≤ 𝑚⁡𝑎𝑛𝑑⁡(𝑘 ≠ 𝑚 ∩ 𝑖 ≠ 𝑛)

{2; 𝑓𝑜𝑟⁡1 ≤ 𝑗 ≤ 𝑚, 𝑘, 𝑖 = 𝑜𝑡ℎ𝑒𝑟𝑠⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑛−2 𝑟(𝑦|𝑊) = {(𝑎𝑖𝑘 ); ⁡1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑘 ≤ 2 }, where 𝑎𝑖𝑘 = {1; 𝑓𝑜𝑟⁡1 ≤ 𝑘 ≤

𝑛𝑚−𝑛 ,𝑖 2

=1

It can be seen that every vertex in 𝑎𝑚𝑎𝑙(𝐹6 , 𝑣, 4) have distinct representation 𝑛𝑚 respect to 𝑊, such that the cardinality of resolvng set in 𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑚) is 2 − 1 or 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑚)) ≤

Marsidi

𝑛𝑚 2

− 1. Thus, we conclude that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑚)) = ⁡

𝑛𝑚 2

− 1. ∎

92

On the Metric Dimension of Some Operation Graphs

Fig 3. The Metric Dimension of Amalgamation of Fan Graph A𝑚𝑎𝑙(𝐹6 , 𝑣 = 𝑦, 3). Theorem 2.4. For 𝑚 ≥ 2, 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) = 2.

the

metric

dimension

of

𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)

is

Proof. The shackle of fan graph, denoted by 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)is a connected graph with vertex set 𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) = {𝑥𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚 + 1} ∪ {𝑦𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚 + 1} and edge set 𝐸(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) = {𝑥𝑗 𝑦𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚 + 1} ∪ {𝑥𝑗 𝑦𝑗+1 ; ⁡1 ≤ 𝑗 ≤ 𝑛} ∪ {𝑥𝑗+1 𝑦𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. The cardinality of vertex set and edge set, respectively are |𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚))| = ⁡2𝑚 + 2 and |𝐸(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚))| = 3𝑚 + 1. The proof that the lower bound of 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) ≥ 2. Based on Proposition 1, that 𝑑𝑖𝑚(𝐺) = 1 if only if 𝐺 ≅ 𝑃𝑛 . The graph 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚) does not isomorphic to path 𝑃𝑛 such that 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) ≥ 2. Furthermore, we proof that the upper bound of 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) ≤ 2, we choose the resolving set 𝑊 = {𝑥1 , 𝑦1 }. The representation of the vertices 𝑣 ∈ 𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) respect to 𝑊 as follows. 𝑟(𝑥𝑗 |𝑊) = ⁡ (𝑗 − 1, 𝑗); 𝑗 ∈ 𝑜𝑑𝑑 𝑟(𝑥𝑗 |𝑊) = ⁡ (𝑗, 𝑗 − 1); 𝑗 ∈ 𝑒𝑣𝑒𝑛

𝑟(𝑦𝑗 |𝑊) = ⁡ (𝑗, 𝑗 − 1); 𝑗 ∈ 𝑜𝑑𝑑 𝑟(𝑦𝑗 |𝑊) = ⁡ (𝑗 − 1, 𝑗); 𝑗 ∈ 𝑒𝑣𝑒𝑛

Vertex 𝑣 ∈ 𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) are distict. So, we have the cardinality of resolving set 𝑊 is |𝑊| = 2. Thus, the upper bound of 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) ≤ 2. It concludes that 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚)) = 2. ∎

Marsidi

93

On the Metric Dimension of Some Operation Graphs

Fig 4. The Metric Dimension of Sℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 3). CONCLUSIONS In this paper, the result show that the local metric dimension of some graph operation such as joint graph 𝑃𝑛 + 𝐶𝑚 , amalgamation of parachute, amalgamation of fan, and 𝑠ℎ𝑎𝑐𝑘(𝐻22 , 𝑒, 𝑚). ACKNOWLEDGMENTS We gratefully acknowledge the support from DRPM KEMENRISTEKDIKTI 2018 indonesia. REFERENCES [1] G. Chartrand, E. Salehi, and P. Zhang, "The partition dimension of a graph," Aequationes Math., vol. 59, pp. 45-54, 2000. [2] G. Chartrand, L. Eroh, and M. A. Johnson, " Resolvability in graphs and the metric dimension of a graph," Discrate Appl. Math., vol. 105, pp. 99-113, 2000. [3] Marsidi, Dafik, I. H. Agustin, and R. Alfarisi, “On the local metric dimension of line graph of special graph,” CAUCHY, vol. 4, no. 3, pp. 125-130, 2016. [4] I. G. Yero, D. Kuziak, and J. S. Rodríguez-Velázquez, “On the metric dimension of corona product graphs,” Computers and Mathematics with Applications, vol. 61, pp. 2793-2798, 2011. [5] H. Fernau, P. Heggernes, P. Hof, D. Meister, and R. Saei, “Computing the metric dimension for chain graphs,” Information Processing Letters, vol. 115, pp. 671-676, 2015. [6] J. Cáceres, C. Hernando, M. Mora, I. M. Pelayo, and M. L. Puertas, “On the metric dimension of infinite graphs,” Discrete Applied Mathematics, vol. 160, pp. 2618-2626, 2012. [7] M. Fehr, S. Gosselin, and O. R. Oellermann, “The metric dimension of cayley digraphs,” Discrete Mathematics, vol. 360, pp. 31-41, 2006. [8] T. K. Maryati, A. N. M. Salman, and E. T. Baskoro, “On H-supermagic labelings for certain shackles and amalgamations of a connected graph,” Utilitas Mathematica, 2010. [9] I. H. Agustin, Dafik, S. Latifah, and R. M. Prihandini, “A super (A,D)-Bm-antimagic total covering of a generalized amalgamation of fan graphs,” CAUCHY, vol. 4, no. 4, pp. 146154, 2017.

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94