Artikel Wahyu Tri Budianto M0114057

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The local metric dimension of starbarbell graph,

K

_m

_⊙

P

_n

_{graph, and M¨}

_{obius ladder graph}

Wahyu Tri Budianto and Tri Atmojo Kusmayadi

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret, Surakarta, Indonesia

E-mail: wahyutri25@gmail.com, tri.atmojo.kusmayadi@gmail.com

Abstract. For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in a nontrivial

connected graph G, the representation of a vertex v of G with respect to W is the n-vector r(v|W) = (d(v, w1), d(v, w2), ..., d(v, wn)). W is a local metric set of Gif r(u|W) ̸=r(v|W)

for every pair of adjacent vertices u, v in G. Local metric set with minimum cardinality is called local metric basis ofGand its cardinality is the local metric dimension ofGand denoted bylmd(G). Starbarbell graph SBm₁,m₂,...,mn is a graph obtained from a star graph Sn and

n complete graphs Kmi by merging one vertex from each Kmi and the ith leaf ofSn, where

mi ≥3, 1 ≤i≤ n, andn ≥2. Km⊙Pn graph is a graph obtained from a complete graph

Kmandmcopies of path graphPn, and then joining by an edge each vertex from theithcopy

of Pn with theith vertex ofKm. M¨obius ladder graphMn is a graph obtained from a cycle

graphCn by connecting every pair of verticesu, vinCn ifd(u, v) =diam(Cn) for n≥5. In

this paper, we determine the local metric dimension of starbarbell graph,Km⊙Pngraph, and

M¨obius ladder graph for even positive integersn≥6.

1. Introduction

In 1975, Slater [9] introduced the concept of metric dimension of a graph, where metric generator was called locating set. Then Harary and Melter [3] independently introduced the same concept in 1976, where metric generator was called resolving set. LetGbe a connected graph with vertex set V(G). A set W ⊂V(G) is called a metric generator of G ifd(u, x)̸=d(v, x) for any pair of vertices u and v in G and some vertices x in W, where d(u, x) is the distance between u and

x. Until now, there are several variations of metric generator, and one of them was introduced by Okamoto et al. [7] in 2010. For an ordered set W ={w1, w2, ..., wn} of n distinct vertices

in a nontrivial connected graph G, the representation of a vertex v of G with respect to W is the n-vector r(v|W) = (d(v, w1), d(v, w2), ..., d(v, wn)). The set W is a local metric set ofG if

r(u|W)̸=r(v|W) for every pair of adjacent vertices uand v inG. The local metric set W with minimum cardinality is called local metric basis of Gand the cardinality of local metric basis of

G is the local metric dimension ofGand denoted by lmd(G).

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local metric dimension of starbarbell graph SBm1,m2,...,mn, Km⊙Pn graph, and M¨obius ladder

graph Mnfor even positive integers n.

2. Main Results

2.1. The Local Metric Dimension of Starbarbell Graph

The starbarbell graph SBm1,m2,...,mn is a graph obtained from a star graph Sn and n complete

graphsKmiby merging one vertex from eachKmiand thei

th_{leaf of}_S

n, wheremi ≥3, 1≤i≤n,

and n≥2. The starbarbell graphSBm1,m2,...,mn can be depicted as in Figure 1.

v

1,m1

u

∑n

i=1(mi−2).

Proof. LetSBm1,m2,...,mn be the starbarbell graph withmi≥3, 1≤i≤n, and n≥2.

(i) We will showlmd(SBm1,m2,...,mn)≥

∑n

i=1(mi−2).

To prove this case, the vertex set V(SBm1,m2,...,mn) is partitioned inton+ 1 sets,

V1 = {v1,1, v1,2, ..., v1,m1},

V2 = {v2,1, v2,2, ..., v2,m2}, ..

.

Vn = {vn,1, vn,2, ..., vn,mn}, and

Vn+1 = {u}.

The cardinality of V(SBm1,m2,...,mn) is

∑n

i=1mi + 1. Assume W ⊆ V(SBm1,m2,...,mn) is

chosen such that |W| < ∑n

i=1(mi −2), then |Wc| = |V(SBm1,m2,...,mn)−W| > 2n+ 1.

Based on pigeonhole principle, there will be at least one partitionVi for 1≤i≤nwhich at

least three elements of Vi are elements of Wc too. Thus, there will be at least two distinct

adjacent vertices vi,j andvi,k, with 2≤j, k≤mi where

d(vi,j, vi,1) = d(vi,k, vi,1) = 1,

d(vi,j, vi,p) = d(vi,k, vi,p) = 1, 2≤p≤mi, j ̸=p̸=k,

d(vi,j, u) = d(vi,k, u) = 2,

d(vi,j, vq,1) = d(vi,k, vq,1) = 3, 1≤q≤n, i̸=q, and

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It means we can not construct W as a local metric set if |W| < ∑n

i=1(mi −2). So

lmd(SBm1,m2,...,mn)≥

∑n

i=1(mi−2).

(ii) AssumeW ={vi,j}where 1≤i≤nand 3≤j≤mi. The cardinality ofW is∑ni=1(mi−2).

Then representations of every vertex ofSBm1,m2,...,mn with respect toW are

r(v1,1|W) = (1,1, ...,1,3,3, ...,3, ...,3,3, ...,3),

r(v1,2|W) = (1,1, ...,1,4,4, ...,4, ...,4,4, ...,4),

.. .

r(v1,m1|W) = (1,1, ...,0,4,4, ...,4, ...,4,4, ...,4),

r(v2,1|W) = (3,3, ...,3,1,1, ...,1, ...,3,3, ...,3),

r(v2,2|W) = (4,4, ...,4,1,1, ...,1, ...,4,4, ...,4),

.. .

r(v2,m2|W) = (4,4, ...,4,1,1, ...,0, ...,4,4, ...,4), ..

.

r(vn,1|W) = (3,3, ...,3,3,3, ...,3, ...,1,1, ...,1),

r(vn,2|W) = (4,4, ...,4,4,4, ...,4, ...,1,1, ...,1),

.. .

r(vn,mn|W) = (4,4, ...,4,4,4, ...,4, ...,1,1, ...,0), and

r(u|W) = (2,2, ...,2,2,2, ...,2, ...,2,2, ...,2).

Every pair of adjacent vertices have distinct representations with respect to W, soW is a local metric basis for starbarbell graphSBm1,m2,...,mn. ⊓⊔

2.2. The Local Metric Dimension of Km⊙Pn Graph

By using the definition from Frucht and Harary [2], the corona product Km ⊙Pn graph is a

graph obtained from a complete graphKm and mcopies of path graphPn, and then joining by

m,3

v

m,n

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Theorem 2.2 For all positive integers m and n,

lmd(Km⊙Pn) =

     

    

1, m=n= 1;

2, m= 1, 2≤n≤5;

m−1, m≥2, n= 1;

m⌊n+2₄ ⌋,

{

m= 1, n≥6;

m≥2, n≥2.

Proof. We consider four cases based on the values ofmand n.

Case 1. m=n= 1.

The corona product K1 ⊙P1 graph is a path with two vertices, so it is easy to say that

lmd(Km⊙Pn) = 1 for m= 1 and n= 1.

Case 2. m= 1 and 2≤n≤5.

The corona product K1 ⊙Pn graph for 2≤ n≤5 is a graph which every single vertex in

K1⊙Pn belongs to some complete graphsK3, solmd(K1⊙Pn)̸= 1. AssumeW ={u1, vk}

with k = 2 for n = 2 and k = 3 for 3 ≤ n ≤ 5, then every pair of adjacent vertices have distinct representations with respect to W. Therefore, W is a local metric basis and

lmd(Km⊙Pn) = 2 for m= 1 and 2≤n≤5.

Case 3. m≥2 and n= 1.

P1 is an empty graph. Then by using the results of Rodr´ıguez-Vel´azquez et al. [8] and

Okamoto et al. [7] we have lmd(Km⊙P1) =lmd(Km) =m−1.

Case 4. m= 1, n≥6 and m≥2,n≥2. (i) m= 1 andn≥6.

By the same reason with Case 2, we have lmd(K1 ⊙Pn) ̸= 1 for n ≥ 6. Assume

W ⊆V(K1⊙Pn) is a non-empty set, then we consider three conditions below.

(a) If v1,1, v1,2, v1,3 ∈/ W then d(v1,1, u1) = (v1,2, u1) = 1 and d(v1,1, v1,r) =

(v1,2, v1,r) = 2, for 4 ≤ r ≤ n, so r(v1,1|W) = r(v1,2|W). We know that v1,1

and v1,2 are adjacent vertices, hence W is not a local metric set. In other words, if

W is a local metric set then at least one of three verticesv1,1,v1,2 orv1,3 belongs

toW.

(b) By the same reason with (a), if W is a local metric set then at least one of three verticesv_1,(n₋₂₎,v_1,(n₋₁₎ orv1,n belongs to W.

(c) Assume a vertex v1,t, for 1 ≤ t ≤ n−4, belongs to W. If all of the vertices

v1,(t+1), v1,(t+2), v1,(t+3), and v1,(t+4) do not belong to W, then d(v1,(t+2), u1) =

(v_1,(t+3), u1) = 1 and d(v1,(t+2), v1,r) = (v1,(t+3), v1,r) = 2, with 1 ≤ r ≤ t or

t+ 5≤r≤n, sor(v1,(t+2)|W) =r(v1,(t+3)|W). We know that v1,(t+2) andv1,(t+3)

are two adjacent vertices, thenW is not a local metric set. In other words, ifW is a local metric set and v1,t ∈W, then at least one of four vertices v1,(t+1),v1,(t+2),

v_1,(t+3), or v_1,(t+4) belongs to W.

Based on three conditions above, the construction ofW such thatW is a local metric basis is by choosing every vertices v1,r, for r ≡ 3 (mod 4) and 1 ≤ r ≤ n, as the

elements of W. Then, if n−rmax = 3, we have to choose one of the vertices v1,(n−2),

v_1,(n₋₁₎, orv1,n as the element ofW. Hence the cardinality ofW is ⌊n+2₄ ⌋.

(ii) m≥2 andn≥2.

By using the results of Rodr´ıguez-Vel´azquez et al. [8], for 2 ≤ n ≤ 5 we have

lmd(Km⊙Pn) =m·(lmd(K1+Pn)−1) =m·(lmd(K1⊙Pn)−1) =m·(2−1) =m.

For 2 ≤ n ≤ 5 we have m = m⌊n+2₄ ⌋. Then for n ≥ 6, again, by using the results of Rodr´ıguez-Vel´azquez et al. [8], we have lmd(Km ⊙Pn) = m ·lmd(K1 +Pn) =

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2.3. The Local Metric Dimension of M¨obius Ladder Graph

Harary and Guy [4] defined the M¨obius ladder graphMnis a graph obtained from a cycle graph

Cn by connecting every pair of vertices u, vinCn ifd(u, v) =diam(Cn) for n≥5. The M¨obius

ladder graph Mn can be depicted as in Figure 3 and Figure 4.

v

1

Figure 3. M¨obius ladder graphMn for even positive integersn

v

___n+7

Figure 4. M¨obius ladder graphMn for odd positive integers n

Theorem 2.3 Let Mn be the M¨obius ladder graph. Then for even positive integers n≥6

lmd(Mn) =

{

1, n≡2(mod4); 2, n≡0(mod4).

Proof. We consider two cases based on the values ofn.

Case 1. n≡2(mod4).

The M¨obius ladder graph Mn for n ≡ 2(mod4) is a bipartite graph. Then by using the

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(ii) Assume W = {v1, v2}, then representations of every vertex of Mn with respect to W

are

r(v1|W) = (0,1),

r(v2|W) = (1,0),

r(v3|W) = r(vn

2+2|W) = (2,1),

r(v4|W) = r(vn

2+3|W) = (3,2), ..

.

r(vn+4

4 |W) = r(v 3n

4 |W) = (

n 4,

n 4 −1),

r(vn+8 4

|W) = r(v3n+4

4

|W) = (n

4,n4),

r(vn+12

4 |W) = r(v 3n+8

4 |W) = (

n

4 −1,n4),

.. .

r(vn

2|W) = r(vn−1|W) = (2,3),

r(vn

2+1|W) = r(vn|W) = (1,2).

Every pair of adjacent vertices have distinct representation, soW is a local metric basis

andlmd(Mn) = 2, forn≡0(mod4). ⊓⊔

3. Conclusion

It can be concluded that the strong metric dimension of a starbarbell graph SBm1,m2,...,mn,

Km⊙Pn graph, and a M¨obius ladder graphMn for even positive integers n≥6 are as stated

in Theorem 2.1, Theorem 2.2, and Theorem 2.3, respectively.

Open Problem : Determine the local metric dimension of M¨obius ladder graph Mn for odd

positive integersn≥5.

Acknowledgments

The authors gratefully acknowledge the support from Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret Surakarta. Then, we wish to thank the referees for their suggestions, which helped to improve the paper.

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