abstrak On The Strong Metric Dimension Of Broken Fan Graph, Starbarbell Graph, And Cm ?K Pn Graph artikel ratih
ON THE STRONG METRIC DIMENSION OF BROKEN FAN
GRAPH, STARBARBELL GRAPH, AND Cm ⊙k Pn GRAPH
Ratih Yunia Mayasari, Tri Atmojo Kusmayadi, Santoso Budi Wiyono
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Universitas Sebelas Maret
Abstract. Let G be a connected graph with vertex set V (G) and edge set E(G). For
every pair of vertices u, v ∈ V (G), the interval I[u, v] between u and v to be the collection
of all vertices that belong to some shortest u−v path. A vertex s ∈ V (G) strongly resolves
two vertices u and v if u belongs to a shortest v − s path or v belongs to a shortest u − s
path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of
G are strongly resolved by some vertex of S. The strong metric basis of G is a strong
resolving set with minimal cardinality. The strong metric dimension sdim(G) of a graph
G is defined as the cardinality of strong metric basis. In this paper we determine the
strong metric dimension of a broken fan graph, starbarbell graph, and Cm ⊙k Pn graph.
Keywords : strong metric dimension, strongly resolved set, broken fan graph, starbarbell
graph, Cm ⊙k Pn graph
1. Introduction
The concept of strong metric dimension was presented by Seb¨o and Tannier [9]
in 2004. Oellermann and Peters-Fransen [8] defined for two vertices u and v in a
connected graph G, the interval I[u, v] between u and v to be collection of all vertices
that belong to some shortest path. A vertex s strongly resolves two vertices u and
v if v ∈ I[u, s] or u ∈ I[v, s]. A set S of vertices in a connected graph G is a strong
resolving set for G if every two vertices of G are strongly resolved by some vertex of
S. The smallest cardinality of a strong resolving set of G is called its strong metric
dimension and is denoted by sdim(G).
Some researchers have investigated the strong metric dimension to some graph
classes. In 2004 Seb¨o and Tannier [9] observed the strong metric dimension of
complete graph Kn , cycle graph Cn , and tree. In 2012, Kuziak et al. [6] observed
the strong metric dimension of corona product graph. In the same year, Kratica et
al. [3] determined the metric dimension of hamming graph Hn,k . Kratica et al. [4]
determined the metric dimension of convex polytope Dn and Tn in 2012 too. In 2013
Yi [10] determined the metric dimension of Pn . Kusmayadi et al. [5] determined
the strong metric dimension of some related wheel graph such as sunflower graph,
t-fold wheel graph, helm graph, and friendship graph. In this paper, we determine
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On The Strong Metric Dimension . . .
R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
the strong metric dimension of a broken fan graph, starbarbell graph, and Cm ⊙k Pn
graph.
2. Strong Metric Dimension
Let G be a connected graph with vertex set V (G), edge set E(G), and S =
{s1 , s2 , . . . , sk } ∈ V (G). Oellermann and Peters-Fransen [8] defined the interval
I[u, v] between u and v to be the collection of all vertices that belong to some
shortest u − v path. A vertex s ∈ S strongly resolves two vertices u and v if u ∈
I[v, s] or v ∈ I[u, s]. A vertex set S of G is a strong resolving set of G if every two
distinct vertices of G are strongly resolved by some vertex of S. The strong metric
basis of G is a strong resolving set with minimal cardinality. The strong metric
dimension of a graph G is defined as the cardinality of strong metric basis denoted
by sdim(G). We often make use of the following lemma and properties about strong
metric dimension given by Kratica et al. [4].
Lemma 2.1. Let u, v ∈ V(G), u ̸= v,
(1) d(w,v) ≤ d(u,v) for each w such that uw ∈ E(G), and
(2) d(u,w) ≤ d(u,v) for each w such that vw ∈ E(G).
Then there does not exist vertex a ∈ V(G), a ̸= u,v that strongly resolves vertices u
and v.
Property 2.1. If S ⊂ V(G) is strong resolving set of graph G, then for every two
vertices u,v ∈ V(G) satisfying conditions 1 and 2 of Lemma 2.1, obtained u ∈ S or
v ∈ S.
Property 2.2. If S ⊂ V(G) is strong resolving set of graph G, then for every two
vertices u, v ∈ V(G) satisfying d(u,v) = diam(G), obtained u ∈ S or v ∈ S.
3. The Strong Metric Dimension of a Broken Fan Graph
Gallian [2] defined the broken fan graph BF (a, b) is a graph with V (BF (a, b)) =
{c} ∪ {v1 , v2 , . . . , va } ∪ {u1 , u2 , . . . , ub } and E(BF (a, b)) = {(c, vi )|i = 1, 2, . . . , a} ∪
{(c, ui )|i = 1, 2, . . . , b} ∪ E(Pa ) ∪ E(Pb ) (a ≥ 2 and b ≥ 2).
Lemma 3.1. For every integer a ≥ 2 and b ≥ 3, if S is a strong resolving set of
broken fan graph BF (a, b) then | S |≥ a + b − 2.
Proof. We prove for every two distinct vertices (vi , ub ) and (uj , ub ). For every i =
{1, 2, . . . , a} so d(vi , ub ) = 2 = diam(BF (a, b)) then by using Property 2.2, vi ∈ S
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
or ub ∈ S and for every j ∈ {1, 2, . . . , b − 2} so d(uj , ub ) = 2 = diam(BF (a, b)) then
using Property 2.2, uj ∈ S or ub ∈ S. Therefore, S contains at least one vertex
from distinct set Xib = {vi , ub } for i ∈ {1, 2, . . . , a} and one vertex from distinct
set Yjb = {uj , ub } for j ∈ {1, 2, . . . , b − 2}. The minimum number of vertices from
distinct set Xib is a and the minimum number of vertices from distinct set Yjb is
b − 2. Therefore, | S |≥ a + b − 2.
Lemma 3.2. For every integer a ≥ 2 and b ≥ 3, a set S = {v1 , v2 , . . . , va , u1 , u2 , . . . ,
ub−2 } is a strong resolving set of broken fan graph BF (a, b).
Proof. We prove that every two distinct vertices u, v ∈ (BF (a, b)) \ S, u ̸= v there
exists a vertex s ∈ S which strongly resolves u and v. There are two pairs of vertices
from V (BF (a, b)) \ S.
(1) A pair of vertices (c, uj ).
For every integer i = {1, 2, . . . , a} and j ∈ {b − 1, b}, d(vi , uj ) = 2 =
diam(BF (a, b)), we obtain the shortest vi − uj path : vi , c, uj . Thus, c ∈
I[vi , uj ].
(2) A pair of vertices (ub−1 , ub )
For j = b − 2, d(uj , ub ) = 2 = diam(BF (a, b)), we obtain the shortest
ub−2 − ub path: ub−2 , ub−1 , ub . Thus, ub−1 ∈ I[uj , ub ].
For every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices BF (a, b) \ S. Thus, S is a strong resolving set
of BF (a, b).
Theorem 3.1. Let BF (a, b) be the broken fan graph, then
{
3,
a = 2 and n = 2;
sdim(BF (a, b)) =
a + b − 2, a ≥ 2 and b ≥ 3.
Proof. There are two cases to determine the strong metric dimension of broken fan
graph.
(1) Case 1 (For a = 2 and b = 2).
By using Theorem from Kusmayadi et al. [5] that sdim(fn ) = 2n−1, so that
sdim(BF (2, 2)) = 3 because of BF (2, 2) ∼
= f2 . Hence, sdim(BF (2, 2)) = 3.
(2) Case 2 (For a ≥ 2 and b ≥ 3).
By using Lemma 3.2 a set S = {v1 , v2 , . . . , va , u1 , u2 , . . . , ub−2 } is strong resolving set of broken fan graph BF (a, b) with a ≥ 2 and b ≥ 3. According
to Lemma 3.1, |S| ≥ a + b − 2, S is strong metric basis of broken fan graph
BF (a, b). Hence, sdim(BF (a, b)) = a + b − 2.
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
4. The Strong Metric Dimension of Starbarbell Graph
Starbarbell graph SBm1 ,m2 ,...,mn is a graph obtained from a star graph Sn and n
complete graph Kmi by merging one vertex from each Kmi and the ith -leaf of Sn ,
where mi ≥ 3, 1 ≤ i ≤ n, and n ≥ 2. The vertices set of starbarbell graph is
c, v1,1 , v1,2 , . . . , v1,m1 , v2,1 , v2,2 , . . . , v2,m2 , . . . , vn,1 , vn,2 , . . . , vn,mn .
Lemma 4.1. For every integer mi ≥ 3 and n ≥ 2, if S is a strong resolving set of
∑
starbarbell graph SBm1 ,m2 ,...,mn then | S | ≥ ni=1 (mi − 1) − 1.
Proof. Let us consider a pair of vertices (vi,j , vk,l ) for every i, k = 1, 2, . . . , n with
vi,j ̸= vk,l and j, l = 2, . . . , mi satisfying both of the conditions of Lemma 2.1.
According to Property 2.1, we obtain vi,j ∈ S or vk,l ∈ S. It means that S contains
one vertex from distinct sets Xi,j,k,l = {vi,j , vk,l }. The minimum number of vertices
∑
∑
from distinct sets Xi,j,k,l is ni=1 (mi −1)−1. Therefore, | S |≥ ni=1 (mi −1)−1.
Lemma 4.2. For every integer mi ≥ 3 and n ≥ 2, a set S = {v1,2 , v1,3 , . . . , v1,m1 , v2,2 ,
v2,3 , . . . , v2,m2 , . . . , vn,2 , vn,3 , . . . , vn,mn −1 } is a strong resolving set of starbarbell graph
SBm1 ,m2 ,...,mn .
Proof. We prove that for every two distinct vertices u, v ∈ V (SBm1 ,m2 ,...,mn )\S, u ̸=
v, there exists a vertex s ∈ S which strongly resolves u and v. There are four possible
pairs of vertices.
(1) A pair of vertices (u, vn,mn ).
For every integer i ∈ {1, 2, . . . , n − 1} and j ∈ {2, 3, . . . , mi }, d(vi,j , vn,mn )= 4
= diam(SBm1 ,m2 ,...,mn ), we obtain the shortest vn,mn −vi,j path: vn,mn , vn,1 , u,
vi,1 , vi,j . Thus, u ∈ I[vn,mn , vi,j ].
(2) A pair of vertices (u, vi,1 ).
For every integer i ∈ {1, 2, . . . , n − 1} and j ∈ {2, 3, . . . , mi }, d(u, vi,j )= 2,
we obtain the shortest u − vi,j path: u, vi,1 , vi,j . Thus, vi,1 ∈ I[u, vi,j ].
For i = n, l ∈ {2, 3, . . . , mi − 1}, d(u, vn,l ) = 2, we obtain the shortest u − vn,l
path: u, vn,1 , vn,l . Thus, vn,1 ∈ I[u, vn,l ].
(3) A pair of vertices (vi,1 , vn,mn ).
For every integer i ∈ {1, 2, . . . , n−1} and j ∈ {2, 3, . . . , mi }, d(vi,j , vn,mn ) = 4
= diam(SBm1 ,m2 ,...,mn ), we obtain the shortest vi,j −vn,mn path: vi,j , vi,1 , u, vn,1 ,
vn,mn . Thus, vi,1 ∈ I[vi,j , vn,mn ].
For i = n, k ∈ {1, 2, . . . , n − 1}, and l ∈ {2, 3, . . . , mi }, d(vk,l , vn,mn ) = 4 =
diam(SBm1 ,m2 ,...,mn ), we obtain the shortest vk,l −vn,mn path: vk,l , vk,1 , u, vn,1 ,
vn,mn . Thus, vn,1 ∈ I[vk,l , vn,mn ].
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(4) A pair of vertices (vi,1 , vk,1 ).
For every integer i, k ∈ {1, 2, . . . , n − 1}, i ̸= k, and l ∈ {2, 3, . . . , mi },
d(vi,1 , vk,l )= 3, we obtain the shortest vi,1 − vk,l path: vi,1 , u, vk,1 , vk,l . Thus,
vk,1 ∈ I[vi,1 , vk,l ].
For k = n and l ∈ {2, 3, . . . , mi − 1}, d(vi,1 , vk,l ) = 3, we obtain the shortest
vi,1 − vk,l path: vi,1 , u, vk,1 , vk,l . Thus, vk,1 ∈ I[vi,1 , vk,l ].
From every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices SBm1 ,m2 ,...,mn \ S. Thus S is a strong resolving
set of starbarbell graph SBm1 ,m2 ,...,mn .
Theorem 4.1. Let SBm1 ,m2 ,...,mn be the starbarbell graph with m ≥ 3 and n ≥ 2.
∑
Then sdim(SBm1 ,m2 ,...,mn ) = ni=1 (mi − 1) − 1.
Proof. By using Lemma 4.2, we have a set S = {v1,2 , v1,3 , . . . , v1,m1 , v2,2 , v2,3 , . . . , v2,m2 ,
. . . , vn,2 , vn,3 , . . . , vn,(mn −1) } is a strong resolving set of SBm1 ,m2 ,...,mn graph with
∑
m ≥ 3 and n ≥ 2. According to Lemma 4.1, | S | ≥ ni=1 (mi − 1) − 1, S is a strong
∑
metric basis of SBm1 ,m2 ,...,mn . Hence, sdim(SBm1 ,m2 ,...,mn ) = ni=1 (mi − 1) − 1.
5. The Strong Metric Dimension of Cm ⊙k Pn Graph
By using the definiton from Frucht and Harary [1], the corona product Cm ⊙ Pn
graph is graph obtained from Cm and Pn by taking one copy of Cm and n copies of
Pn and joining by an edge each vertex from ith -copy of Pn with the ith -vertex of Cm .
Then, by using the definiton from Marbun and Salman [7], the k-multilevel corona
product Cm ⊙k Pn graph is graph obtained from the corona product Cm ⊙k−1 Pn and
Pn graph and it can be written as Cm ⊙k Pn = (Cm ⊙k−1 Pn ) ⊙ Pn . The C3 ⊙2 P2
can be depicted as in Figure 1.
v1,2,2 v1,2,1 v1,1,2 v1,1,1
v1,1
v1,0,2
v1,2
v1
v1,0,1
v2,0,1
v2,0,2
v3
v2,1
v3,2
v2
v2,1,1
v2,1,2
v2,2,1
v3,0,2
v3,2,2
v3,2,1
v3,0,1
v2,2
v3,1
v2,2,2
v3,1,2
v3,1,1
Figure 1. C3 ⊙2 P2 graph
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
Lemma 5.1. For every integer m ≥ 3, n = 2, and k ≥ 1, if S is a strong resolving
set of Cm ⊙k Pn graph then | S |≥ (mn(n + 1)k−1 ) − 1.
Proof. Let us consider a pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ) with y, z = k + 1,
1 ≤ a1 , b1 ≤ m, 0 ≤ ai , bi ≤ 2, and 2 ≤ i ≤ k + 1 satisfying both of the conditions of
Lemma 2.1. According to Property 2.1, we obtain va1 ,a2 ,...,ay ∈ S or vb1 ,b2 ,...,bz ∈ S.
It means that S contains one vertex from distinct sets Xyz = {va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz }.
The minimum number of vertices from distinct sets Xyz is (mn(n + 1)k+1 ) − 1.
Therefore, | S |≥ (mn(n + 1)k+1 ) − 1.
Lemma 5.2. For every integer m ≥ 3, n = 2, and k ≥ 1, a set S = {v1,0,...,1 , v1,0,...,2 ,
. . . , v2,0,...,1 , v2,0,...,2 , . . . , vm,2,...,1 } is a strong resolving set of Cm ⊙k Pn graph.
Proof. We prove that every two distinct vertices u, v ∈ (Cm ⊙k Pn ) \ S, there exists
a vertex s ∈ S which strongly resolves u and v. There are two pairs of vertices from
V (Cm ⊙k Pn ) \ S.
(1) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ).
For every integer 1 ≤ y, z ≤ k, 1 ≤ a1 , b1 ≤ m, 0 ≤ ai , bi ≤ 2, and
2 ≤ i ≤ k, we obtain the shortest va1 ,a2 ,...,ay − vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 path:
va1 ,a2 ,...,ay , va1 ,a2 ,...,ay−1 , . . . , va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz , vb1 ,b2 ,...,bz ,bz+1 ,
. . . , vb1 ,b2 ,...,bz ,,bz+1 ,...,bk+1 . So that, vb1 ,b2 ,...,bz ∈ I[va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 ].
(2) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ).
For every integer 1 ≤ y ≤ k, 1 ≤ a1 ≤ m − 1, 0 ≤ ai ≤ 2, and 2 ≤ i ≤ k,
z = k + 1, b1 = m, bj = 2, and 2 ≤ j ≤ k + 1, we obtain the shortest
va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 − vb1 ,b2 ,...,bz path: va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , . . . , va1 ,a2 ,...,ay , . . . ,
va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz . So that, va1 ,a2 ,...,ay ∈ I[va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , vb1 ,b2 ,...,bz ].
For every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices (Cm ⊙k Pn ) \ S. Thus S is a strong resolving set
of Cm ⊙k Pn .
Lemma 5.3. For every integer m ≥ 3, n ≥ 3, and k ≥ 1, if S is a strong resolving
set of Cm ⊙k Pn graph then | S |≥ (mn(n + 1)k−1 ) − 2.
Proof. We know that S is strong resolving set of Cm ⊙k Pn graph. Suppose that S
contains at most (mn(n + 1)k−1 ) − 3 vertices, then |S| < (mn(n + 1)k−1 ) − 2. Let V1
is set of vertices va1 ,a2 ,...,ay with y = k + 1, 1 ≤ a1 ≤ m, 0 ≤ ai ≤ 2, and 2 ≤ i ≤ k
and V2 is set of vertices vb1 ,b2 ,...,bz with 1 ≤ z ≤ k, 1 ≤ b1 ≤ m, 0 ≤ bi ≤ 2, and
2 ≤ i ≤ k. Now, we define S1 = V1 ∩ S and S2 = V2 ∩ S. Without loss of generality,
we may take S1 = p, p > 0 and S2 = q, q ≥ 0. Clearly p + q ≤ (mn(n + 1)k−1 ) − 3,
there are two distinct vertices va and vb where va ∈ V1 \ S and vb ∈ V2 \ S such that
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
for every s ∈ S we obtain va ∈
/ I[vb , s] and vb ∈
/ I[va , s]. This contradicts with the
supposition that S is a strong resolving set. Thus |S| ≥ (mn(n + 1)k−1 ) − 2.
Lemma 5.4. For every integer m ≥ 3, n ≥ 3, and k ≥ 1, a set S = {v1,0,...,1 , v1,0,...,2 ,
. . . , v1,0,...,n , . . . , v2,0,...,1 , v2,0,...,2 , . . . , v2,0,...,n , . . . , vm,n,...,1 , vm,n,...,2 , . . . , vm,n,...,n−2 } is a
strong resolving set of Cm ⊙k Pn graph.
Proof. We prove that every two distinct vertices u, v ∈ (Cm ⊙k Pn ) \ S, there exists
a vertex s ∈ S which strongly resolves u and v. There are three pairs of vertices
from V (Cm ⊙k Pn ) \ S.
(1) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ).
For every integer 1 ≤ y, z ≤ k, 1 ≤ a1 , b1 ≤ m, 0 ≤ ai , bi ≤ n, and
2 ≤ i ≤ k, we obtain the shortest va1 ,a2 ,...,ay − vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 path:
va1 ,a2 ,...,ay , va1 ,a2 ,...,ay−1 , . . . , va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz , vb1 ,b2 ,...,bz ,bz+1 , . . . ,
vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 . So that, vb1 ,b2 ,...,bz ∈ I[va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 ].
(2) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz )
For every integer 1 ≤ y ≤ k, 1 ≤ a1 ≤ m − 1, 0 ≤ ai ≤ n, 2 ≤ i ≤
k, z = k + 1, b1 = m, bj = n, 2 ≤ j ≤ k, bz = {n − 1, n} we obtain
the shortest va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 − vb1 ,b2 ,...,bz path: va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , . . . ,
va1 ,a2 ,...,ay , . . . , va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz . So that, va1 ,a2 ,...,ay ∈
I[va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , vb1 ,b2 ,...,bz ].
(3) A pair of vertices (va1 ,a2 ,...,ay , va1 ,a2 ,...,az )
For every integer x, y, z = k + 1, a1 = m, ai = n, 2 ≤ i ≤ k, ay = n −
1, az = n ax = n − 2 we obtain the shortest va1 ,a2 ,...,ax − va1 ,a2 ,...,az path:
va1 ,a2 ,...,ax , va1 ,a2 ,...,ay , va1 ,a2 ,...,az . So that, va1 ,a2 ,...,ay ∈ I[va1 ,a2 ,...,ax , va1 ,a2 ,...,az ].
For every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices (Cm ⊙k Pn ) \ S. Thus S is a strong resolving set
of Cm ⊙k Pn .
Theorem 5.1. Let Cm ⊙k Pn be the corona product of cycle graph and path graph,
then
{
(mn(n + 1)k−1 ) − 1, m ≥ 3, k ≥ 1, dan n = 2;
sdim(Cm ⊙k Pn ) =
(mn(n + 1)k−1 ) − 2, m ≥ 3, k ≥ 1, dan n ≥ 3.
Proof. By Lemma 5.1 and Lemma 5.2, we have sdim(Cm ⊙k Pn ) = (mn(n+1)k−1 )−1
for m ≥ 3, n = 2, and k ≥ 1. By Lemma 5.3 and Lemma 5.4, we have sdim(Cm ⊙k
Pn ) = (mn(n + 1)k−1 ) − 2 for m ≥ 3, n ≥ 3, and k ≥ 1.
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
6. Conclusion
According to the discussion above it can be concluded that the strong metric
dimension of a broken fan graph BF (a, b), a starbarbell graph SBm1 ,m2 ,...,mn , and
a Cm ⊙k Pn graph are as stated in Theorem 3.1, Theorem 4.1, and Theorem 5.1,
respectively.
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8
2018
GRAPH, STARBARBELL GRAPH, AND Cm ⊙k Pn GRAPH
Ratih Yunia Mayasari, Tri Atmojo Kusmayadi, Santoso Budi Wiyono
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Universitas Sebelas Maret
Abstract. Let G be a connected graph with vertex set V (G) and edge set E(G). For
every pair of vertices u, v ∈ V (G), the interval I[u, v] between u and v to be the collection
of all vertices that belong to some shortest u−v path. A vertex s ∈ V (G) strongly resolves
two vertices u and v if u belongs to a shortest v − s path or v belongs to a shortest u − s
path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of
G are strongly resolved by some vertex of S. The strong metric basis of G is a strong
resolving set with minimal cardinality. The strong metric dimension sdim(G) of a graph
G is defined as the cardinality of strong metric basis. In this paper we determine the
strong metric dimension of a broken fan graph, starbarbell graph, and Cm ⊙k Pn graph.
Keywords : strong metric dimension, strongly resolved set, broken fan graph, starbarbell
graph, Cm ⊙k Pn graph
1. Introduction
The concept of strong metric dimension was presented by Seb¨o and Tannier [9]
in 2004. Oellermann and Peters-Fransen [8] defined for two vertices u and v in a
connected graph G, the interval I[u, v] between u and v to be collection of all vertices
that belong to some shortest path. A vertex s strongly resolves two vertices u and
v if v ∈ I[u, s] or u ∈ I[v, s]. A set S of vertices in a connected graph G is a strong
resolving set for G if every two vertices of G are strongly resolved by some vertex of
S. The smallest cardinality of a strong resolving set of G is called its strong metric
dimension and is denoted by sdim(G).
Some researchers have investigated the strong metric dimension to some graph
classes. In 2004 Seb¨o and Tannier [9] observed the strong metric dimension of
complete graph Kn , cycle graph Cn , and tree. In 2012, Kuziak et al. [6] observed
the strong metric dimension of corona product graph. In the same year, Kratica et
al. [3] determined the metric dimension of hamming graph Hn,k . Kratica et al. [4]
determined the metric dimension of convex polytope Dn and Tn in 2012 too. In 2013
Yi [10] determined the metric dimension of Pn . Kusmayadi et al. [5] determined
the strong metric dimension of some related wheel graph such as sunflower graph,
t-fold wheel graph, helm graph, and friendship graph. In this paper, we determine
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On The Strong Metric Dimension . . .
R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
the strong metric dimension of a broken fan graph, starbarbell graph, and Cm ⊙k Pn
graph.
2. Strong Metric Dimension
Let G be a connected graph with vertex set V (G), edge set E(G), and S =
{s1 , s2 , . . . , sk } ∈ V (G). Oellermann and Peters-Fransen [8] defined the interval
I[u, v] between u and v to be the collection of all vertices that belong to some
shortest u − v path. A vertex s ∈ S strongly resolves two vertices u and v if u ∈
I[v, s] or v ∈ I[u, s]. A vertex set S of G is a strong resolving set of G if every two
distinct vertices of G are strongly resolved by some vertex of S. The strong metric
basis of G is a strong resolving set with minimal cardinality. The strong metric
dimension of a graph G is defined as the cardinality of strong metric basis denoted
by sdim(G). We often make use of the following lemma and properties about strong
metric dimension given by Kratica et al. [4].
Lemma 2.1. Let u, v ∈ V(G), u ̸= v,
(1) d(w,v) ≤ d(u,v) for each w such that uw ∈ E(G), and
(2) d(u,w) ≤ d(u,v) for each w such that vw ∈ E(G).
Then there does not exist vertex a ∈ V(G), a ̸= u,v that strongly resolves vertices u
and v.
Property 2.1. If S ⊂ V(G) is strong resolving set of graph G, then for every two
vertices u,v ∈ V(G) satisfying conditions 1 and 2 of Lemma 2.1, obtained u ∈ S or
v ∈ S.
Property 2.2. If S ⊂ V(G) is strong resolving set of graph G, then for every two
vertices u, v ∈ V(G) satisfying d(u,v) = diam(G), obtained u ∈ S or v ∈ S.
3. The Strong Metric Dimension of a Broken Fan Graph
Gallian [2] defined the broken fan graph BF (a, b) is a graph with V (BF (a, b)) =
{c} ∪ {v1 , v2 , . . . , va } ∪ {u1 , u2 , . . . , ub } and E(BF (a, b)) = {(c, vi )|i = 1, 2, . . . , a} ∪
{(c, ui )|i = 1, 2, . . . , b} ∪ E(Pa ) ∪ E(Pb ) (a ≥ 2 and b ≥ 2).
Lemma 3.1. For every integer a ≥ 2 and b ≥ 3, if S is a strong resolving set of
broken fan graph BF (a, b) then | S |≥ a + b − 2.
Proof. We prove for every two distinct vertices (vi , ub ) and (uj , ub ). For every i =
{1, 2, . . . , a} so d(vi , ub ) = 2 = diam(BF (a, b)) then by using Property 2.2, vi ∈ S
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
or ub ∈ S and for every j ∈ {1, 2, . . . , b − 2} so d(uj , ub ) = 2 = diam(BF (a, b)) then
using Property 2.2, uj ∈ S or ub ∈ S. Therefore, S contains at least one vertex
from distinct set Xib = {vi , ub } for i ∈ {1, 2, . . . , a} and one vertex from distinct
set Yjb = {uj , ub } for j ∈ {1, 2, . . . , b − 2}. The minimum number of vertices from
distinct set Xib is a and the minimum number of vertices from distinct set Yjb is
b − 2. Therefore, | S |≥ a + b − 2.
Lemma 3.2. For every integer a ≥ 2 and b ≥ 3, a set S = {v1 , v2 , . . . , va , u1 , u2 , . . . ,
ub−2 } is a strong resolving set of broken fan graph BF (a, b).
Proof. We prove that every two distinct vertices u, v ∈ (BF (a, b)) \ S, u ̸= v there
exists a vertex s ∈ S which strongly resolves u and v. There are two pairs of vertices
from V (BF (a, b)) \ S.
(1) A pair of vertices (c, uj ).
For every integer i = {1, 2, . . . , a} and j ∈ {b − 1, b}, d(vi , uj ) = 2 =
diam(BF (a, b)), we obtain the shortest vi − uj path : vi , c, uj . Thus, c ∈
I[vi , uj ].
(2) A pair of vertices (ub−1 , ub )
For j = b − 2, d(uj , ub ) = 2 = diam(BF (a, b)), we obtain the shortest
ub−2 − ub path: ub−2 , ub−1 , ub . Thus, ub−1 ∈ I[uj , ub ].
For every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices BF (a, b) \ S. Thus, S is a strong resolving set
of BF (a, b).
Theorem 3.1. Let BF (a, b) be the broken fan graph, then
{
3,
a = 2 and n = 2;
sdim(BF (a, b)) =
a + b − 2, a ≥ 2 and b ≥ 3.
Proof. There are two cases to determine the strong metric dimension of broken fan
graph.
(1) Case 1 (For a = 2 and b = 2).
By using Theorem from Kusmayadi et al. [5] that sdim(fn ) = 2n−1, so that
sdim(BF (2, 2)) = 3 because of BF (2, 2) ∼
= f2 . Hence, sdim(BF (2, 2)) = 3.
(2) Case 2 (For a ≥ 2 and b ≥ 3).
By using Lemma 3.2 a set S = {v1 , v2 , . . . , va , u1 , u2 , . . . , ub−2 } is strong resolving set of broken fan graph BF (a, b) with a ≥ 2 and b ≥ 3. According
to Lemma 3.1, |S| ≥ a + b − 2, S is strong metric basis of broken fan graph
BF (a, b). Hence, sdim(BF (a, b)) = a + b − 2.
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
4. The Strong Metric Dimension of Starbarbell Graph
Starbarbell graph SBm1 ,m2 ,...,mn is a graph obtained from a star graph Sn and n
complete graph Kmi by merging one vertex from each Kmi and the ith -leaf of Sn ,
where mi ≥ 3, 1 ≤ i ≤ n, and n ≥ 2. The vertices set of starbarbell graph is
c, v1,1 , v1,2 , . . . , v1,m1 , v2,1 , v2,2 , . . . , v2,m2 , . . . , vn,1 , vn,2 , . . . , vn,mn .
Lemma 4.1. For every integer mi ≥ 3 and n ≥ 2, if S is a strong resolving set of
∑
starbarbell graph SBm1 ,m2 ,...,mn then | S | ≥ ni=1 (mi − 1) − 1.
Proof. Let us consider a pair of vertices (vi,j , vk,l ) for every i, k = 1, 2, . . . , n with
vi,j ̸= vk,l and j, l = 2, . . . , mi satisfying both of the conditions of Lemma 2.1.
According to Property 2.1, we obtain vi,j ∈ S or vk,l ∈ S. It means that S contains
one vertex from distinct sets Xi,j,k,l = {vi,j , vk,l }. The minimum number of vertices
∑
∑
from distinct sets Xi,j,k,l is ni=1 (mi −1)−1. Therefore, | S |≥ ni=1 (mi −1)−1.
Lemma 4.2. For every integer mi ≥ 3 and n ≥ 2, a set S = {v1,2 , v1,3 , . . . , v1,m1 , v2,2 ,
v2,3 , . . . , v2,m2 , . . . , vn,2 , vn,3 , . . . , vn,mn −1 } is a strong resolving set of starbarbell graph
SBm1 ,m2 ,...,mn .
Proof. We prove that for every two distinct vertices u, v ∈ V (SBm1 ,m2 ,...,mn )\S, u ̸=
v, there exists a vertex s ∈ S which strongly resolves u and v. There are four possible
pairs of vertices.
(1) A pair of vertices (u, vn,mn ).
For every integer i ∈ {1, 2, . . . , n − 1} and j ∈ {2, 3, . . . , mi }, d(vi,j , vn,mn )= 4
= diam(SBm1 ,m2 ,...,mn ), we obtain the shortest vn,mn −vi,j path: vn,mn , vn,1 , u,
vi,1 , vi,j . Thus, u ∈ I[vn,mn , vi,j ].
(2) A pair of vertices (u, vi,1 ).
For every integer i ∈ {1, 2, . . . , n − 1} and j ∈ {2, 3, . . . , mi }, d(u, vi,j )= 2,
we obtain the shortest u − vi,j path: u, vi,1 , vi,j . Thus, vi,1 ∈ I[u, vi,j ].
For i = n, l ∈ {2, 3, . . . , mi − 1}, d(u, vn,l ) = 2, we obtain the shortest u − vn,l
path: u, vn,1 , vn,l . Thus, vn,1 ∈ I[u, vn,l ].
(3) A pair of vertices (vi,1 , vn,mn ).
For every integer i ∈ {1, 2, . . . , n−1} and j ∈ {2, 3, . . . , mi }, d(vi,j , vn,mn ) = 4
= diam(SBm1 ,m2 ,...,mn ), we obtain the shortest vi,j −vn,mn path: vi,j , vi,1 , u, vn,1 ,
vn,mn . Thus, vi,1 ∈ I[vi,j , vn,mn ].
For i = n, k ∈ {1, 2, . . . , n − 1}, and l ∈ {2, 3, . . . , mi }, d(vk,l , vn,mn ) = 4 =
diam(SBm1 ,m2 ,...,mn ), we obtain the shortest vk,l −vn,mn path: vk,l , vk,1 , u, vn,1 ,
vn,mn . Thus, vn,1 ∈ I[vk,l , vn,mn ].
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(4) A pair of vertices (vi,1 , vk,1 ).
For every integer i, k ∈ {1, 2, . . . , n − 1}, i ̸= k, and l ∈ {2, 3, . . . , mi },
d(vi,1 , vk,l )= 3, we obtain the shortest vi,1 − vk,l path: vi,1 , u, vk,1 , vk,l . Thus,
vk,1 ∈ I[vi,1 , vk,l ].
For k = n and l ∈ {2, 3, . . . , mi − 1}, d(vi,1 , vk,l ) = 3, we obtain the shortest
vi,1 − vk,l path: vi,1 , u, vk,1 , vk,l . Thus, vk,1 ∈ I[vi,1 , vk,l ].
From every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices SBm1 ,m2 ,...,mn \ S. Thus S is a strong resolving
set of starbarbell graph SBm1 ,m2 ,...,mn .
Theorem 4.1. Let SBm1 ,m2 ,...,mn be the starbarbell graph with m ≥ 3 and n ≥ 2.
∑
Then sdim(SBm1 ,m2 ,...,mn ) = ni=1 (mi − 1) − 1.
Proof. By using Lemma 4.2, we have a set S = {v1,2 , v1,3 , . . . , v1,m1 , v2,2 , v2,3 , . . . , v2,m2 ,
. . . , vn,2 , vn,3 , . . . , vn,(mn −1) } is a strong resolving set of SBm1 ,m2 ,...,mn graph with
∑
m ≥ 3 and n ≥ 2. According to Lemma 4.1, | S | ≥ ni=1 (mi − 1) − 1, S is a strong
∑
metric basis of SBm1 ,m2 ,...,mn . Hence, sdim(SBm1 ,m2 ,...,mn ) = ni=1 (mi − 1) − 1.
5. The Strong Metric Dimension of Cm ⊙k Pn Graph
By using the definiton from Frucht and Harary [1], the corona product Cm ⊙ Pn
graph is graph obtained from Cm and Pn by taking one copy of Cm and n copies of
Pn and joining by an edge each vertex from ith -copy of Pn with the ith -vertex of Cm .
Then, by using the definiton from Marbun and Salman [7], the k-multilevel corona
product Cm ⊙k Pn graph is graph obtained from the corona product Cm ⊙k−1 Pn and
Pn graph and it can be written as Cm ⊙k Pn = (Cm ⊙k−1 Pn ) ⊙ Pn . The C3 ⊙2 P2
can be depicted as in Figure 1.
v1,2,2 v1,2,1 v1,1,2 v1,1,1
v1,1
v1,0,2
v1,2
v1
v1,0,1
v2,0,1
v2,0,2
v3
v2,1
v3,2
v2
v2,1,1
v2,1,2
v2,2,1
v3,0,2
v3,2,2
v3,2,1
v3,0,1
v2,2
v3,1
v2,2,2
v3,1,2
v3,1,1
Figure 1. C3 ⊙2 P2 graph
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
Lemma 5.1. For every integer m ≥ 3, n = 2, and k ≥ 1, if S is a strong resolving
set of Cm ⊙k Pn graph then | S |≥ (mn(n + 1)k−1 ) − 1.
Proof. Let us consider a pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ) with y, z = k + 1,
1 ≤ a1 , b1 ≤ m, 0 ≤ ai , bi ≤ 2, and 2 ≤ i ≤ k + 1 satisfying both of the conditions of
Lemma 2.1. According to Property 2.1, we obtain va1 ,a2 ,...,ay ∈ S or vb1 ,b2 ,...,bz ∈ S.
It means that S contains one vertex from distinct sets Xyz = {va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz }.
The minimum number of vertices from distinct sets Xyz is (mn(n + 1)k+1 ) − 1.
Therefore, | S |≥ (mn(n + 1)k+1 ) − 1.
Lemma 5.2. For every integer m ≥ 3, n = 2, and k ≥ 1, a set S = {v1,0,...,1 , v1,0,...,2 ,
. . . , v2,0,...,1 , v2,0,...,2 , . . . , vm,2,...,1 } is a strong resolving set of Cm ⊙k Pn graph.
Proof. We prove that every two distinct vertices u, v ∈ (Cm ⊙k Pn ) \ S, there exists
a vertex s ∈ S which strongly resolves u and v. There are two pairs of vertices from
V (Cm ⊙k Pn ) \ S.
(1) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ).
For every integer 1 ≤ y, z ≤ k, 1 ≤ a1 , b1 ≤ m, 0 ≤ ai , bi ≤ 2, and
2 ≤ i ≤ k, we obtain the shortest va1 ,a2 ,...,ay − vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 path:
va1 ,a2 ,...,ay , va1 ,a2 ,...,ay−1 , . . . , va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz , vb1 ,b2 ,...,bz ,bz+1 ,
. . . , vb1 ,b2 ,...,bz ,,bz+1 ,...,bk+1 . So that, vb1 ,b2 ,...,bz ∈ I[va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 ].
(2) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ).
For every integer 1 ≤ y ≤ k, 1 ≤ a1 ≤ m − 1, 0 ≤ ai ≤ 2, and 2 ≤ i ≤ k,
z = k + 1, b1 = m, bj = 2, and 2 ≤ j ≤ k + 1, we obtain the shortest
va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 − vb1 ,b2 ,...,bz path: va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , . . . , va1 ,a2 ,...,ay , . . . ,
va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz . So that, va1 ,a2 ,...,ay ∈ I[va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , vb1 ,b2 ,...,bz ].
For every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices (Cm ⊙k Pn ) \ S. Thus S is a strong resolving set
of Cm ⊙k Pn .
Lemma 5.3. For every integer m ≥ 3, n ≥ 3, and k ≥ 1, if S is a strong resolving
set of Cm ⊙k Pn graph then | S |≥ (mn(n + 1)k−1 ) − 2.
Proof. We know that S is strong resolving set of Cm ⊙k Pn graph. Suppose that S
contains at most (mn(n + 1)k−1 ) − 3 vertices, then |S| < (mn(n + 1)k−1 ) − 2. Let V1
is set of vertices va1 ,a2 ,...,ay with y = k + 1, 1 ≤ a1 ≤ m, 0 ≤ ai ≤ 2, and 2 ≤ i ≤ k
and V2 is set of vertices vb1 ,b2 ,...,bz with 1 ≤ z ≤ k, 1 ≤ b1 ≤ m, 0 ≤ bi ≤ 2, and
2 ≤ i ≤ k. Now, we define S1 = V1 ∩ S and S2 = V2 ∩ S. Without loss of generality,
we may take S1 = p, p > 0 and S2 = q, q ≥ 0. Clearly p + q ≤ (mn(n + 1)k−1 ) − 3,
there are two distinct vertices va and vb where va ∈ V1 \ S and vb ∈ V2 \ S such that
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for every s ∈ S we obtain va ∈
/ I[vb , s] and vb ∈
/ I[va , s]. This contradicts with the
supposition that S is a strong resolving set. Thus |S| ≥ (mn(n + 1)k−1 ) − 2.
Lemma 5.4. For every integer m ≥ 3, n ≥ 3, and k ≥ 1, a set S = {v1,0,...,1 , v1,0,...,2 ,
. . . , v1,0,...,n , . . . , v2,0,...,1 , v2,0,...,2 , . . . , v2,0,...,n , . . . , vm,n,...,1 , vm,n,...,2 , . . . , vm,n,...,n−2 } is a
strong resolving set of Cm ⊙k Pn graph.
Proof. We prove that every two distinct vertices u, v ∈ (Cm ⊙k Pn ) \ S, there exists
a vertex s ∈ S which strongly resolves u and v. There are three pairs of vertices
from V (Cm ⊙k Pn ) \ S.
(1) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ).
For every integer 1 ≤ y, z ≤ k, 1 ≤ a1 , b1 ≤ m, 0 ≤ ai , bi ≤ n, and
2 ≤ i ≤ k, we obtain the shortest va1 ,a2 ,...,ay − vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 path:
va1 ,a2 ,...,ay , va1 ,a2 ,...,ay−1 , . . . , va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz , vb1 ,b2 ,...,bz ,bz+1 , . . . ,
vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 . So that, vb1 ,b2 ,...,bz ∈ I[va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz ,bz+1 ,...,bk+1 ].
(2) A pair of vertices (va1 ,a2 ,...,ay , vb1 ,b2 ,...,bz )
For every integer 1 ≤ y ≤ k, 1 ≤ a1 ≤ m − 1, 0 ≤ ai ≤ n, 2 ≤ i ≤
k, z = k + 1, b1 = m, bj = n, 2 ≤ j ≤ k, bz = {n − 1, n} we obtain
the shortest va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 − vb1 ,b2 ,...,bz path: va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , . . . ,
va1 ,a2 ,...,ay , . . . , va1 , . . . , vb1 , . . . , vb1 ,b2 ,...,bz . So that, va1 ,a2 ,...,ay ∈
I[va1 ,a2 ,...,ay ,ay+1 ,...,ak+1 , vb1 ,b2 ,...,bz ].
(3) A pair of vertices (va1 ,a2 ,...,ay , va1 ,a2 ,...,az )
For every integer x, y, z = k + 1, a1 = m, ai = n, 2 ≤ i ≤ k, ay = n −
1, az = n ax = n − 2 we obtain the shortest va1 ,a2 ,...,ax − va1 ,a2 ,...,az path:
va1 ,a2 ,...,ax , va1 ,a2 ,...,ay , va1 ,a2 ,...,az . So that, va1 ,a2 ,...,ay ∈ I[va1 ,a2 ,...,ax , va1 ,a2 ,...,az ].
For every possible pairs of vertices, there exists a vertex s ∈ S which strongly
resolves every two distinct vertices (Cm ⊙k Pn ) \ S. Thus S is a strong resolving set
of Cm ⊙k Pn .
Theorem 5.1. Let Cm ⊙k Pn be the corona product of cycle graph and path graph,
then
{
(mn(n + 1)k−1 ) − 1, m ≥ 3, k ≥ 1, dan n = 2;
sdim(Cm ⊙k Pn ) =
(mn(n + 1)k−1 ) − 2, m ≥ 3, k ≥ 1, dan n ≥ 3.
Proof. By Lemma 5.1 and Lemma 5.2, we have sdim(Cm ⊙k Pn ) = (mn(n+1)k−1 )−1
for m ≥ 3, n = 2, and k ≥ 1. By Lemma 5.3 and Lemma 5.4, we have sdim(Cm ⊙k
Pn ) = (mn(n + 1)k−1 ) − 2 for m ≥ 3, n ≥ 3, and k ≥ 1.
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R. Y. Mayasari, T. A. Kusmayadi, S. B. Wiyono
6. Conclusion
According to the discussion above it can be concluded that the strong metric
dimension of a broken fan graph BF (a, b), a starbarbell graph SBm1 ,m2 ,...,mn , and
a Cm ⊙k Pn graph are as stated in Theorem 3.1, Theorem 4.1, and Theorem 5.1,
respectively.
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