ARTIKEL MILA WIDYANINGRUM M0114028
On The Strong Metric Dimension of Sun Graph,
Windmill Graph, and M¨
obius Ladder Graph
Mila Widyaningrum and Tri Atmojo Kusmayadi
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas
Sebelas Maret, Surakarta, Indonesia
E-mail: [email protected], [email protected]
Abstract. Let G be a connected graph with vertex set V (G) and edge set E(G). The interval
I[u, v] between u and v is defined as 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 set S ⊂ V (G) is a strong resolving set if every
two distinct vertices of G are strongly resolved by some vertex of S. The smallest cardinality of
strong resolving set is called a strong metric basis. The strong metric dimension of G, denoted
by sdim(G), is defined as the cardinality of the strong metric basis. In this paper we determine
obius ladder
the strong metric dimension of a sun graph Sn , a windmill graph Knm , and a M¨
graph Mn . We obtain the strong metric dimension of sun graph Sn is n - 1 for n ≥ 3. The
strong metric dimension of windmill graph Knm is (n - 1)m - 1 for m ≥ 2 and n ≥ 3. The strong
⌉ for n even.
metric dimension of M¨
obius ladder graph Mn with n ≥ 5 is 2⌈ n+2
4
1. Introduction
Seb¨
o and Tannier [8] introduced the concept of strong metric dimension in 2004. In 2007 the
strong metric dimension of graph and digraph was introduced by Oellermann and Peters-Fransen
[7]. Let G be a connected graph with vertex set V (G) and edge set E(G), Oellermann and PetersFransen [7] defined for two 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. If u belongs to a shortest v-s
path, denoted by u ∈ I[v, s], or v belongs to a shortest u-s path, denoted by v ∈ I[u, s] then a
vertex s strongly resolves two vertices u and v. A set of vertices S ⊂ V (G) is strong resolving
set for G if every two distinct vertices of G are strongly resolved by some vertex of S. The
smallest cardinality of strong resolving set is called strong metric basis. The cardinality of the
strong metric basis is a strong metric dimension of G, denoted by sdim(G).
Some authors have investigated the strong metric dimension to some graph classes. In 2004
Seb¨
o and Tannier [8] observed the strong metric dimension of complete graph Kn , cycle graph
Cn , and tree. Kratica [3] determined the strong metric dimension of hamming graph Hn,k in
2012. At the same year, Kratica [4] already observed the strong metric dimension of some convex
polytope graph. In 2013 Yi [9] determined that the metric dimension of G is 1 if and only if G
is a path Pn and the strong metric dimension of G is n - 1 if and only if G is complete graph
Kn . 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 2016. In this
paper we determine the strong metric dimension of a sun graph Sn , a windmill graph Knm , and
a M¨obius Ladder graph Mn for n even.
2. Main Results
2.1. Strong Metric Dimension
Let G be a connected graph with a set of vertices V (G) and a set of edges E(G). Let S be a
subset of V (G). Oellermann and Peters-Fransen [7] defined for two 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 ∈ 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 smallest cardinality of strong resolving set is called strong
metric basis of G. 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,
(i) d(w,v) ≤ d(u,v) for each w such that uw ∈ E(G), and
(ii) 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.
2.2. The Strong Metric Dimension of Sun Graph
Mirzakhah and Kiani [6] defined the sun graph Sn for n ≥ 3 as a graph obtained by connecting
each vertex of cycle Cn to a pendant vertex with an edge. The sun graph Sn can be depicted as
in Figure 1.
v2
u2
v1
u1
u3
un
vn
v3
u4
v4
Figure 1. Sun graph Sn
Theorem 2.1 Let Sn be the sun graph with n ≥ 3. Then sdim(Sn ) = n − 1.
Proof. First, we prove that for every integer n ≥ 3, if S is a strong resolving set of sun graph Sn
then | S | ≥ n − 1. We know that S is a strong resolving set of sun graph Sn . Suppose that S
contains at most n - 2 vertices, then | S | < n - 1. Let V1 , V2 ⊂ V (Sn ), with V1 = {v1 , v2 , . . . , vn }
and V2 = {u1 , u2 , . . . , un }. Now, we define S1 = V1 ∩ S and S2 = V2 ∩ S. Without loss of
generality, we may take | S1 | = a, a > 0 and | S2 | = b, b ≥ 0. We obtain a + b ≤ n - 2, there
are two distinct vertices vi and vj ∈ V1 \ S1 , where i ̸= j such that for every s ∈ S, vi ∈
/ I[vj , s]
and vj ∈
/ I[vi , s]. This contradicts with the supposition that S is a strong resolving set. Thus
| S | ≥ n - 1.
Then, we prove that for every integer n ≥ 3, if S = {v1 , v2 , . . . , vn−1 } then S is a strong
resolving set of sun graph Sn . We prove that for every two distinct vertices u, v ∈ V (Sn ) \ S
where u ̸= v is strongly resolved by a vertex s ∈ S. There are two possible pairs of vertices.
(i) A pair of vertices (ui , vn ) with i = 1, 2, . . ., n.
For every integer 1 ≤ i ≤ ⌈ n−1
⌉ and
2 ⌉, we obtain the shortest path between v⌈ n−1
2
vn is v⌈ n−1 ⌉ , u⌈ n−1 ⌉ , u⌈ n−1 ⌉−1 , . . . , u1 , un , vn . So that ui ∈ I[v⌈ n−1 ⌉ , vn ]. Then for
2
2
2
2
every integer ⌊ n+1
⌋ and vn is
2 ⌋ ≤ i ≤ n, we obtain the shortest path between v⌊ n+1
2
v⌊ n+1 ⌋ , u⌊ n+1 ⌋ , u⌊ n+1 ⌋+1 , . . . , un−1 , un , vn . So that ui ∈ I[v⌊ n+1 ⌋ , vn ].
2
2
2
2
(ii) A pair of vertices (ui , uj ) with i, j = 1, 2, . . ., n and i ̸= j.
For every integer i, j ∈ [1, n] without loss of generality we may assume 1 ≤ i < j ≤ n,
therefore ui and uj will be strongly resolved by vi so that ui ∈ I[vi , uj ].
From every possible pairs of vertices, there exists a vertex vi ∈ S with i ∈ [1, n − 1] which
strongly resolves two distinct vertices of V (Sn ) \ S. Thus S is a strong resolving set of sun
graph Sn .
We obtain a set S = {v1 , v2 , . . . , vn−1 } is a strong resolving set of sun graph Sn with n ≥ 3
and we have | S | ≥ n - 1, so that S is a strong metric basis of sun graph Sn . Hence sdim(Sn )
= n - 1.
⊓
⊔
2.3. The Strong Metric Dimension of Windmill Graph
Gallian [1] defined the windmill graph Knm is a graph consists of m copies of Kn with a vertex
in common. The windmill graph Knm can be depicted as in Figure 2.
v2
v1
vn - 2
vn - 1
v(n -1)m
v(n -1)m - 1
vn
c
v(n -1)(m -1) + 2
v(n -1)(m -1) + 1
v(n -1) 2
vn + 1
v2n - 3
Figure 2. Windmill graph Knm
Theorem 2.2 Let Knm be the windmill graph with m ≥ 2 and n ≥ 3. Then sdim(Knm ) =
(n − 1)m − 1.
Proof. First, we prove that for every integer m ≥ 2 and n ≥ 3, if S is a strong resolving set of
windmill graph Knm then | S | ≥ (n - 1)m - 1. Consider a pair of vertices (vi , vj ) for i, j = 1, 2,
. . ., (n − 1)m and i ̸= j satisfying conditions 1 and 2 of Lemma 2.1. According to Property 2.1,
we obtain vi ∈ S or vj ∈ S. It means S has at least one vertex from sets Xij = {vi , vj } with i, j
= 1, 2, . . ., (n − 1)m and i ̸= j. The minimum number of vertices from sets Xij is (n - 1)m - 1.
Therefore, | S | ≥ (n − 1)m−1.
After that, we prove that for every integer m ≥ 2 and n ≥ 3, if S = {v1 , v2 , . . . , v(n−1)m−1 }
then S is a strong resolving set of Knm graph. For every two distinct vertices u, v ∈ V (Knm )\S,
there is a vertex s ∈ S which strongly resolves u and v. Let us prove that a pair of vertices
(c, v(n−1)m ) is strongly resolved by vi for i = 1, 2, . . ., (n − 1)(m − 1). The shortest path between
v(n−1)m and vi is v(n−1)m , c, vi . Since vertex c belongs to this shortest path, vi strongly resolves
c and v(n−1)m . Thus S is a strong resolving set of windmill graph Knm .
We have a set S = {v1 , v2 , . . . , v(n−1)m−1 } is a strong resolving set of windmill graph Knm and
|S| ≥ (n - 1)m - 1, so that S is a strong metric basis of windmill graph Knm . Hence sdim(Knm ) =
(n - 1)m - 1.
⊓
⊔
2.4. The Strong Metric Dimension of M¨
obius Ladder Graph
Guy and Harary [2] defined the M¨obius ladder graph Mn is a graph constructed by connecting
vertices u and v in the cycle Cn if d(u,v) = diam(Cn ). The M¨obius ladder graph Mn can be
depicted as in Figure 3 and Figure 4.
v1
vn
v1
v2
v_n +4
v3
2
v_n +3
vn_ +1
vn_
2
v4
2
v_n +2
2
v_n +1
2
vn
v2
vn_ +2
2
vn_+ 4 vn_+3
2
2
v_n
v3
v4
2
(a)
2
Figure 3. M¨obius ladder graph Mn for n even
v1
vn
v1
v2
v__
n+1
v__
n+7
v3
2
2
v__
n+3
v__
n+5
v__
n+7
2
v__
n+3
2
2
2
v4
v__
n+5
v2
2
vn
v4
v__
n+1
2
v3
(b)
Figure 4. M¨obius ladder graph Mn for n odd
Theorem 2.3 Let Mn be the M¨
obius ladder graph with n ≥ 5. Then sdim(Mn ) = 2 ⌈ n+2
4 ⌉ for
n even.
Proof. First, we prove that for every n ≥ 5, if S is a strong resolving set of M¨obius ladder graph
obius ladder
Mn then | S | ≥ 2 ⌈ n+2
4 ⌉ for n even. We know that S is a strong resolving set of M¨
n+2
graph Mn . Suppose that S contains at most 2 ⌈ n+2
4 ⌉ − 1 vertices, then | S | < 2 ⌈ 4 ⌉. Let
V1 , V2 ⊂ V (Mn ), with V1 = {v1 , v2 , . . ., v n2 } and V2 = {v n2 +1 , v n2 +2 , . . . , vn }. 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 ≥ 2 ⌈ n+2
4 ⌉. Otherwise, there are two distinct vertices vi and vj , where
vi ∈ V1 \S1 and vj ∈ V2 \S2 such that for every s ∈ S we obtain vi ∈
/ I[vj , s] and vj ∈
/ I[vi , s].
This contradicts with the supposition that S is a strong resolving set. Thus | S | ≥ 2 ⌈ n+2
4 ⌉.
Then, we prove that for every even integer n ≥ 5, a set S = {v1 , v2 , . . ., v⌈ n+2 ⌉ , v n2 +1 , v n2 +2 ,
4
. . ., v n +⌈ n+2 ⌉ } is a strong resolving set of M¨obius ladder graph Mn for n even. We prove that for
2
4
n
n
n+2
every two distinct vertices vi , vj ∈ V (Mn )\S, where i, j = (⌈ n+2
4 ⌉, 2 ] ∪ ( 2 +⌈ 4 ⌉, n] is strongly
resolved by a vertex s ∈ S. There are three possible pairs of vertices.
n
(i) A pair of vertices (vi , vj ) where ⌈ n+2
4 ⌉ < i, j ≤ 2 , i ̸= j.
n
n+2
n
For every integer ⌈ n+2
4 ⌉ < i, j ≤ 2 with ⌈ 4 ⌉ < i < j ≤ 2 , we obtain the shortest path
between v⌈ n+2 ⌉ and vj is v⌈ n+2 ⌉ , v⌈ n+2 ⌉+1 , . . ., vi , . . ., vj . So that vi ∈ I[v⌈ n+2 ⌉ , vj ].
4
4
4
4
(ii) A pair of vertices (vi , vj ) where n2 +⌈ n+2
4 ⌉ < i, j ≤ n, i ̸= j.
n
n+2
For every integer 2 +⌈ 4 ⌉ < i, j ≤ n with n2 +⌈ n+2
4 ⌉ < i < j ≤ n, we obtain the
shortest path between v n +⌈ n+2 ⌉ and vj is v n +⌈ n+2 ⌉ , v n +⌈ n+2 ⌉+1 , . . . , vi , . . ., vj . So that
2
4
2
4
2
4
vi ∈ I[v n +⌈ n+2 ⌉ , vj ].
2
4
n
n
n+2
(iii) A pair of vertices (vi , vj ) where ⌈ n+2
4 ⌉ < i ≤ 2 and 2 + ⌈ 4 ⌉ < j ≤ n.
n
n
n+2
For every pair of vertices (vi , vj ) with ⌈ 4 ⌉ < i ≤ 2 and 2 + ⌈ n+2
4 ⌉ < j ≤ n. Vertex vi and
n
n+2
vj will be strongly resolved by a vertex v n +⌈ n+2 ⌉ if 2 + ⌈ 4 ⌉ < j ≤ i + n2 , we obtain the
2
4
shortest path between vi and v n +⌈ n+2 ⌉ is vi , vi+ n2 , vi+ n2 −1 , vi+ n2 −2 , . . . , v n +⌈ n+2 ⌉ . So that
2
4
2
4
vj ∈ I[vi , v n +⌈ n+2 ⌉ ]. If i + n2 ≤ j ≤ n, then vi and vj will be strongly resolved by v1 with
2
4
the shortest path between vi and v1 is vi , vi+ n2 , vi+ n2 +1 , . . . , vn , v1 . So that vj ∈ I[vi , v1 ].
From every possible pairs of vertices, there exists a vertex s ∈ S which strongly resolves two
distinct vertices of V (Mn ) \ S. Thus S is a strong resolving set of M¨obius ladder graph Mn .
We obtain a set S = {v1 , v2 , . . . , v⌈ n+2 ⌉ , v n2 +1 , v n2 +2 , . . . , v n +⌈ n+2 ⌉ } is a strong resolving set
4
2
4
of M¨obius ladder graph Mn and we have | S | ≥ 2 ⌈ n+2
4 ⌉, so that S is a strong metric basis of
n+2
Mn . Hence sdim(Mn ) = 2 ⌈ 4 ⌉ for n even.
⊓
⊔
3. Conclusion
According to the discussion above, we have the strong metric dimension of a sun graph Sn ,
a windmill graph Knm , and a M¨obius ladder graph Mn for n even. Furthermore, we conclude
this paper with the following open problem for the direction of further research which is still in
progress.
Open Problem: Determine the strong metric dimension of a M¨obius ladder graph Mn for n
odd.
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.
References
[1] Gallian, Joseph A., A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics (2016),
♯(DS6).
[2] Guy, R. and F. Harary, On The M¨
obius Ladders, Canad. Math. Bull 10 (1967), no. 4, 493–496.
ˇ
[3] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. Cangalovi´
c, Minimal Doubly Resolving Sets and The Strong Metric
Dimension of Hamming Graphs, Applicable Analysis and Discrete Mathematics 6 (2012), 63–71.
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[4] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. Cangalovi´
c, Minimal Doubly Resolving Sets and The Strong Metric
Dimension of Some Convex Polytope, Applied Mathematics and Computation 218 (2012), 9790–9801.
[5] Kusmayadi, T. A., S. Kuntari, D. Rahmadi, and F. A. Lathifah, On The Strong Metric Dimension of Some
Related Wheel Graph, Far East Journal of Mathematical Sciences (FJMS) 99 (2016), no. 9, 1325–1334.
[6] Mirzakhah, M. and D. Kiani, The Sun Graph is Determined by Its Signless Laplacian Spectrum, Electronic
Journal of Linear Algebra 20 (2010), 610–620.
[7] Oellermann, O. and J. Peters-Fransen, The Strong Metric Dimension of Graphs and Digraphs, Discrete Applied
Mathematics 155 (2007), 356–364.
[8] Seb¨
o, A. and E. Tannier, On Metric Generators of Graph, Mathematics and Operations Research 29(2)
(2004), 383–393.
[9] Yi, E., On Strong Metric Dimension Graphs and Their Complements, Acta Mathematica Sinica 29(8) (2013),
1479–1492.
Windmill Graph, and M¨
obius Ladder Graph
Mila Widyaningrum and Tri Atmojo Kusmayadi
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas
Sebelas Maret, Surakarta, Indonesia
E-mail: [email protected], [email protected]
Abstract. Let G be a connected graph with vertex set V (G) and edge set E(G). The interval
I[u, v] between u and v is defined as 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 set S ⊂ V (G) is a strong resolving set if every
two distinct vertices of G are strongly resolved by some vertex of S. The smallest cardinality of
strong resolving set is called a strong metric basis. The strong metric dimension of G, denoted
by sdim(G), is defined as the cardinality of the strong metric basis. In this paper we determine
obius ladder
the strong metric dimension of a sun graph Sn , a windmill graph Knm , and a M¨
graph Mn . We obtain the strong metric dimension of sun graph Sn is n - 1 for n ≥ 3. The
strong metric dimension of windmill graph Knm is (n - 1)m - 1 for m ≥ 2 and n ≥ 3. The strong
⌉ for n even.
metric dimension of M¨
obius ladder graph Mn with n ≥ 5 is 2⌈ n+2
4
1. Introduction
Seb¨
o and Tannier [8] introduced the concept of strong metric dimension in 2004. In 2007 the
strong metric dimension of graph and digraph was introduced by Oellermann and Peters-Fransen
[7]. Let G be a connected graph with vertex set V (G) and edge set E(G), Oellermann and PetersFransen [7] defined for two 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. If u belongs to a shortest v-s
path, denoted by u ∈ I[v, s], or v belongs to a shortest u-s path, denoted by v ∈ I[u, s] then a
vertex s strongly resolves two vertices u and v. A set of vertices S ⊂ V (G) is strong resolving
set for G if every two distinct vertices of G are strongly resolved by some vertex of S. The
smallest cardinality of strong resolving set is called strong metric basis. The cardinality of the
strong metric basis is a strong metric dimension of G, denoted by sdim(G).
Some authors have investigated the strong metric dimension to some graph classes. In 2004
Seb¨
o and Tannier [8] observed the strong metric dimension of complete graph Kn , cycle graph
Cn , and tree. Kratica [3] determined the strong metric dimension of hamming graph Hn,k in
2012. At the same year, Kratica [4] already observed the strong metric dimension of some convex
polytope graph. In 2013 Yi [9] determined that the metric dimension of G is 1 if and only if G
is a path Pn and the strong metric dimension of G is n - 1 if and only if G is complete graph
Kn . 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 2016. In this
paper we determine the strong metric dimension of a sun graph Sn , a windmill graph Knm , and
a M¨obius Ladder graph Mn for n even.
2. Main Results
2.1. Strong Metric Dimension
Let G be a connected graph with a set of vertices V (G) and a set of edges E(G). Let S be a
subset of V (G). Oellermann and Peters-Fransen [7] defined for two 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 ∈ 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 smallest cardinality of strong resolving set is called strong
metric basis of G. 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,
(i) d(w,v) ≤ d(u,v) for each w such that uw ∈ E(G), and
(ii) 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.
2.2. The Strong Metric Dimension of Sun Graph
Mirzakhah and Kiani [6] defined the sun graph Sn for n ≥ 3 as a graph obtained by connecting
each vertex of cycle Cn to a pendant vertex with an edge. The sun graph Sn can be depicted as
in Figure 1.
v2
u2
v1
u1
u3
un
vn
v3
u4
v4
Figure 1. Sun graph Sn
Theorem 2.1 Let Sn be the sun graph with n ≥ 3. Then sdim(Sn ) = n − 1.
Proof. First, we prove that for every integer n ≥ 3, if S is a strong resolving set of sun graph Sn
then | S | ≥ n − 1. We know that S is a strong resolving set of sun graph Sn . Suppose that S
contains at most n - 2 vertices, then | S | < n - 1. Let V1 , V2 ⊂ V (Sn ), with V1 = {v1 , v2 , . . . , vn }
and V2 = {u1 , u2 , . . . , un }. Now, we define S1 = V1 ∩ S and S2 = V2 ∩ S. Without loss of
generality, we may take | S1 | = a, a > 0 and | S2 | = b, b ≥ 0. We obtain a + b ≤ n - 2, there
are two distinct vertices vi and vj ∈ V1 \ S1 , where i ̸= j such that for every s ∈ S, vi ∈
/ I[vj , s]
and vj ∈
/ I[vi , s]. This contradicts with the supposition that S is a strong resolving set. Thus
| S | ≥ n - 1.
Then, we prove that for every integer n ≥ 3, if S = {v1 , v2 , . . . , vn−1 } then S is a strong
resolving set of sun graph Sn . We prove that for every two distinct vertices u, v ∈ V (Sn ) \ S
where u ̸= v is strongly resolved by a vertex s ∈ S. There are two possible pairs of vertices.
(i) A pair of vertices (ui , vn ) with i = 1, 2, . . ., n.
For every integer 1 ≤ i ≤ ⌈ n−1
⌉ and
2 ⌉, we obtain the shortest path between v⌈ n−1
2
vn is v⌈ n−1 ⌉ , u⌈ n−1 ⌉ , u⌈ n−1 ⌉−1 , . . . , u1 , un , vn . So that ui ∈ I[v⌈ n−1 ⌉ , vn ]. Then for
2
2
2
2
every integer ⌊ n+1
⌋ and vn is
2 ⌋ ≤ i ≤ n, we obtain the shortest path between v⌊ n+1
2
v⌊ n+1 ⌋ , u⌊ n+1 ⌋ , u⌊ n+1 ⌋+1 , . . . , un−1 , un , vn . So that ui ∈ I[v⌊ n+1 ⌋ , vn ].
2
2
2
2
(ii) A pair of vertices (ui , uj ) with i, j = 1, 2, . . ., n and i ̸= j.
For every integer i, j ∈ [1, n] without loss of generality we may assume 1 ≤ i < j ≤ n,
therefore ui and uj will be strongly resolved by vi so that ui ∈ I[vi , uj ].
From every possible pairs of vertices, there exists a vertex vi ∈ S with i ∈ [1, n − 1] which
strongly resolves two distinct vertices of V (Sn ) \ S. Thus S is a strong resolving set of sun
graph Sn .
We obtain a set S = {v1 , v2 , . . . , vn−1 } is a strong resolving set of sun graph Sn with n ≥ 3
and we have | S | ≥ n - 1, so that S is a strong metric basis of sun graph Sn . Hence sdim(Sn )
= n - 1.
⊓
⊔
2.3. The Strong Metric Dimension of Windmill Graph
Gallian [1] defined the windmill graph Knm is a graph consists of m copies of Kn with a vertex
in common. The windmill graph Knm can be depicted as in Figure 2.
v2
v1
vn - 2
vn - 1
v(n -1)m
v(n -1)m - 1
vn
c
v(n -1)(m -1) + 2
v(n -1)(m -1) + 1
v(n -1) 2
vn + 1
v2n - 3
Figure 2. Windmill graph Knm
Theorem 2.2 Let Knm be the windmill graph with m ≥ 2 and n ≥ 3. Then sdim(Knm ) =
(n − 1)m − 1.
Proof. First, we prove that for every integer m ≥ 2 and n ≥ 3, if S is a strong resolving set of
windmill graph Knm then | S | ≥ (n - 1)m - 1. Consider a pair of vertices (vi , vj ) for i, j = 1, 2,
. . ., (n − 1)m and i ̸= j satisfying conditions 1 and 2 of Lemma 2.1. According to Property 2.1,
we obtain vi ∈ S or vj ∈ S. It means S has at least one vertex from sets Xij = {vi , vj } with i, j
= 1, 2, . . ., (n − 1)m and i ̸= j. The minimum number of vertices from sets Xij is (n - 1)m - 1.
Therefore, | S | ≥ (n − 1)m−1.
After that, we prove that for every integer m ≥ 2 and n ≥ 3, if S = {v1 , v2 , . . . , v(n−1)m−1 }
then S is a strong resolving set of Knm graph. For every two distinct vertices u, v ∈ V (Knm )\S,
there is a vertex s ∈ S which strongly resolves u and v. Let us prove that a pair of vertices
(c, v(n−1)m ) is strongly resolved by vi for i = 1, 2, . . ., (n − 1)(m − 1). The shortest path between
v(n−1)m and vi is v(n−1)m , c, vi . Since vertex c belongs to this shortest path, vi strongly resolves
c and v(n−1)m . Thus S is a strong resolving set of windmill graph Knm .
We have a set S = {v1 , v2 , . . . , v(n−1)m−1 } is a strong resolving set of windmill graph Knm and
|S| ≥ (n - 1)m - 1, so that S is a strong metric basis of windmill graph Knm . Hence sdim(Knm ) =
(n - 1)m - 1.
⊓
⊔
2.4. The Strong Metric Dimension of M¨
obius Ladder Graph
Guy and Harary [2] defined the M¨obius ladder graph Mn is a graph constructed by connecting
vertices u and v in the cycle Cn if d(u,v) = diam(Cn ). The M¨obius ladder graph Mn can be
depicted as in Figure 3 and Figure 4.
v1
vn
v1
v2
v_n +4
v3
2
v_n +3
vn_ +1
vn_
2
v4
2
v_n +2
2
v_n +1
2
vn
v2
vn_ +2
2
vn_+ 4 vn_+3
2
2
v_n
v3
v4
2
(a)
2
Figure 3. M¨obius ladder graph Mn for n even
v1
vn
v1
v2
v__
n+1
v__
n+7
v3
2
2
v__
n+3
v__
n+5
v__
n+7
2
v__
n+3
2
2
2
v4
v__
n+5
v2
2
vn
v4
v__
n+1
2
v3
(b)
Figure 4. M¨obius ladder graph Mn for n odd
Theorem 2.3 Let Mn be the M¨
obius ladder graph with n ≥ 5. Then sdim(Mn ) = 2 ⌈ n+2
4 ⌉ for
n even.
Proof. First, we prove that for every n ≥ 5, if S is a strong resolving set of M¨obius ladder graph
obius ladder
Mn then | S | ≥ 2 ⌈ n+2
4 ⌉ for n even. We know that S is a strong resolving set of M¨
n+2
graph Mn . Suppose that S contains at most 2 ⌈ n+2
4 ⌉ − 1 vertices, then | S | < 2 ⌈ 4 ⌉. Let
V1 , V2 ⊂ V (Mn ), with V1 = {v1 , v2 , . . ., v n2 } and V2 = {v n2 +1 , v n2 +2 , . . . , vn }. 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 ≥ 2 ⌈ n+2
4 ⌉. Otherwise, there are two distinct vertices vi and vj , where
vi ∈ V1 \S1 and vj ∈ V2 \S2 such that for every s ∈ S we obtain vi ∈
/ I[vj , s] and vj ∈
/ I[vi , s].
This contradicts with the supposition that S is a strong resolving set. Thus | S | ≥ 2 ⌈ n+2
4 ⌉.
Then, we prove that for every even integer n ≥ 5, a set S = {v1 , v2 , . . ., v⌈ n+2 ⌉ , v n2 +1 , v n2 +2 ,
4
. . ., v n +⌈ n+2 ⌉ } is a strong resolving set of M¨obius ladder graph Mn for n even. We prove that for
2
4
n
n
n+2
every two distinct vertices vi , vj ∈ V (Mn )\S, where i, j = (⌈ n+2
4 ⌉, 2 ] ∪ ( 2 +⌈ 4 ⌉, n] is strongly
resolved by a vertex s ∈ S. There are three possible pairs of vertices.
n
(i) A pair of vertices (vi , vj ) where ⌈ n+2
4 ⌉ < i, j ≤ 2 , i ̸= j.
n
n+2
n
For every integer ⌈ n+2
4 ⌉ < i, j ≤ 2 with ⌈ 4 ⌉ < i < j ≤ 2 , we obtain the shortest path
between v⌈ n+2 ⌉ and vj is v⌈ n+2 ⌉ , v⌈ n+2 ⌉+1 , . . ., vi , . . ., vj . So that vi ∈ I[v⌈ n+2 ⌉ , vj ].
4
4
4
4
(ii) A pair of vertices (vi , vj ) where n2 +⌈ n+2
4 ⌉ < i, j ≤ n, i ̸= j.
n
n+2
For every integer 2 +⌈ 4 ⌉ < i, j ≤ n with n2 +⌈ n+2
4 ⌉ < i < j ≤ n, we obtain the
shortest path between v n +⌈ n+2 ⌉ and vj is v n +⌈ n+2 ⌉ , v n +⌈ n+2 ⌉+1 , . . . , vi , . . ., vj . So that
2
4
2
4
2
4
vi ∈ I[v n +⌈ n+2 ⌉ , vj ].
2
4
n
n
n+2
(iii) A pair of vertices (vi , vj ) where ⌈ n+2
4 ⌉ < i ≤ 2 and 2 + ⌈ 4 ⌉ < j ≤ n.
n
n
n+2
For every pair of vertices (vi , vj ) with ⌈ 4 ⌉ < i ≤ 2 and 2 + ⌈ n+2
4 ⌉ < j ≤ n. Vertex vi and
n
n+2
vj will be strongly resolved by a vertex v n +⌈ n+2 ⌉ if 2 + ⌈ 4 ⌉ < j ≤ i + n2 , we obtain the
2
4
shortest path between vi and v n +⌈ n+2 ⌉ is vi , vi+ n2 , vi+ n2 −1 , vi+ n2 −2 , . . . , v n +⌈ n+2 ⌉ . So that
2
4
2
4
vj ∈ I[vi , v n +⌈ n+2 ⌉ ]. If i + n2 ≤ j ≤ n, then vi and vj will be strongly resolved by v1 with
2
4
the shortest path between vi and v1 is vi , vi+ n2 , vi+ n2 +1 , . . . , vn , v1 . So that vj ∈ I[vi , v1 ].
From every possible pairs of vertices, there exists a vertex s ∈ S which strongly resolves two
distinct vertices of V (Mn ) \ S. Thus S is a strong resolving set of M¨obius ladder graph Mn .
We obtain a set S = {v1 , v2 , . . . , v⌈ n+2 ⌉ , v n2 +1 , v n2 +2 , . . . , v n +⌈ n+2 ⌉ } is a strong resolving set
4
2
4
of M¨obius ladder graph Mn and we have | S | ≥ 2 ⌈ n+2
4 ⌉, so that S is a strong metric basis of
n+2
Mn . Hence sdim(Mn ) = 2 ⌈ 4 ⌉ for n even.
⊓
⊔
3. Conclusion
According to the discussion above, we have the strong metric dimension of a sun graph Sn ,
a windmill graph Knm , and a M¨obius ladder graph Mn for n even. Furthermore, we conclude
this paper with the following open problem for the direction of further research which is still in
progress.
Open Problem: Determine the strong metric dimension of a M¨obius ladder graph Mn for n
odd.
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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