artikel Tiffani Arzaqi Putri M0113050
ON THE STRONG METRIC DIMENSION OF A LOLLIPOP
GRAPH, A GENERALIZED WEB GRAPH, AND A
GENERALIZED FLOWER GRAPH
Tiffani Arzaqi Putri and Tri Atmojo Kusmayadi
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Sebelas Maret University
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 to be the collection of all vertices that belong to some
shortest u − v path. A vertex s strongly resolves two vertices u and v 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]. 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 lollipop Lm,n graph, generalized web W B(G, m, n) graph,
and generalized flower F L(G, m, n, p) graph.
Keywords : strong metric dimension, strong resolving, lollipop graph, generalized web
graph, generalized flower graph
1. Introduction
Let G be a simple connected undirected graph G = (V, E), where V is a set of
vertices, and E is a set of edges. The distance between vertices u and v, i.e. the
length of a shortest u-v path is denoted by d(u, v). A vertex set B={x1 , x2 , . . . , xk }
of G is a resolving set of G if every two distinct vertices of G are resolved by some
vertex of B. The metric basis of G is a resolving set with minimal cardinality. The
metric dimension of G, denoted by dim(G) is the cardinality of its metric basis.
The concept of strong metric dimension was introduced by Seb¨o and Tannier [6]
in 2004. Kratica et al. [2] defined for two vertices u and v in a connected graph
G, 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 u,v ∈ V (G) are strongly resolved by some vertex of S. The
strong metric dimension of G denoted by sdim(G) is the minimum cardinality over
all strong resolving sets of G.
Many researchers have investigated the strong metric dimension to some graph
classes. In 2004 Seb¨o and Tannier [6] observed that the strong metric dimension of
complete graph Kn is n - 1, cycle graph Cn is ⌈ n2 ⌉, and tree is L(T ) - 1, where L(T )
1
On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
denotes the number of leaves of tree. In 2013 Yi [8] determined that the strong
metric dimension of G is 1 if and only if G = Pn . Kusmayadi et al. [3] determined
the strong metric dimension of some related wheel graph such as sunflower graph, tfold wheel graph, helm graph, and friendship graph. In this paper, we determine the
strong metric dimension of a lollipop Lm,n graph, a generalized web W B(G, m, n)
graph, and a generalized flower F L(G, m, n, p) graph.
2. Main Result
2.1. 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). Oelermann and PetersFransen [5] 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. [1].
Lemma 2.1. Let u, v ∈ V(G), u ̸= v,
(1) d(w,v) ≤ d(u,v) for each w such that {u,w} ∈ E(G), and
(2) d(u,w) ≤ d(u,v) for each w such that {v,w} ∈ 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 a strong resolving set of graph G, then for every two
vertices u,v ∈ V(G) which satisfy conditions 1 and 2 of Lemma 2.1, obtained u ∈ S
or v ∈ S.
We denote diam(G) as the diameter of graph G, i.e. the maximal distance between
two vertices in G.
Property 2.2. If S ⊂ V(G) is a strong resolving set of graph G, then for every two
vertices u, v ∈ V(G) such that d(u,v) = diam(G), obtained u ∈ S or v ∈ S.
2.2. The Strong Metric Dimension of a Lollipop Graph. Weisstein [7] defined
the lollipop graph Lm,n for m ≥ 3, n ≥ 2 as a graph obtained by joining a complete
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
graph Km to a path graph Pn with a bridge. We assume that the lollipop graph has
a vertex set V (Lm,n )={v1 , v2 , . . . , vm , u1 , u2 , . . . , un }. The lollipop graph Lm,n can
be depicted as in Figure 1.
Figure 1. Lollipop graph Lm,n
Lemma 2.2. For every integer m ≥ 3 and n ≥ 1, if S is a strong resolving set of
lollipop graph Lm,n then | S | ≥ m − 1.
Proof. Consider a pair of vertex (vi , un ) with i ∈ [1,m-1] satisfying condition d(vi , un )
= n+1 = diam(Lm,n ). According to Property 2.2, we obtain vi ∈ S or un ∈ S,
therefore S has at least m − 1 vertices. Hence | S | ≥ m − 1.
Lemma 2.3. For every integer m ≥ 3 and n ≥ 1, a set S = {v1 , v2 , . . . , vm−1 } is a
strong resolving set of lollipop graph Lm,n .
Proof. For every integer i ∈ [1, m-1] satisfying condition d(un , vi ) = n+1 = diam(Lm,n ),
we obtain the shortest un − vi path: un , un−1 , un−2 , . . . , u1 , vm , vi . So that vi strongly
resolves a pair of vertices (vm ,un ). Thus vm ∈ I[un , vi ].
For a pair of vertices (uk , ul ) with k ̸= l, k, l ∈ [1, n-1], without loss of generality k
< l we obtain for i ∈ [1,m-1], the shortest ul − vi path: ul , ul−1 , . . . , uk , uk−1 , . . . , u1 ,
vm , vi . So that vi strongly resolves a pair of vertices (uk , ul ). Thus uk ∈ I[ul , vi ].
Therefore S = {v1 , v2 , . . . , vm−1 } is a strong resolving set of lollipop graph Lm,n .
Theorem 2.1. Let Lm,n be the lollipop graph with m ≥ 3 and n ≥ 1. Then
sdim(Lm,n ) = m − 1.
Proof. By using Lemma 2.3 a set S = {v1 , v2 , . . . , vm−1 } is a strong resolving set of
lollipop graph Lm,n with m ≥ 3 and n ≥ 1. According to Lemma 2.2, | S | ≥ m - 1
so that S is a strong metric basis of lollipop graph Lm,n . Hence sdim(Lm,n ) = m 1.
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T. A. Putri, T. A. Kusmayadi
2.3. The Strong Metric Dimension of a Generalized Web Graph. Miller
et al. [4] defined the generalized web graph W B(G, m, n) for m ≥ 3, n ≥ 2 is
the graph obtained from the generalized pyramid P (G, m) by taking p copies of
Pn (n ≥ 2) and merging an end vertex of a different copy of Pn with each vertex
of the furthermost copy of G from the apex. In this paper we obtain the strong
metric dimension of generalized web graph with G ∼
= Cm and without center vertex
denoted by W B0 (Cm , n). We assume that the generalized web graph has a vertex
1
2
n−1 n n
n
set V (W Bo )={v11 , v21 , . . . , vm
, v12 , v22 , . . . , vm
, . . . , v1n−1 , v2n−1 , . . . , vm
, v 1 , v 2 , . . . , vm
}.
The generalized web graph W B0 (Cm , n) can be depicted as in Figure 2.
Figure 2. Generalized web graph W B0 (Cm , n)
Lemma 2.4. For every integer m ≥ 3 and n ≥ 2, if S is a strong resolving set of
generalized web graph W B0 (Cm , n) then | S | ≥ m.
Proof. Let S is a strong resolving set of generalized web graph W B0 (Cm , n). Suppose
that S contains at most m - 1 vertices, then | S | < m. Let V1 , V2 ⊂ V (W B0 (Cm , n)),
1
2
n−1
, v12 , v22 , . . . , vm
, . . . , v1n−1 , v2n−1 , . . . , vm
} and V2 = {v1n , v2n ,
with V1 = {v11 , v21 , . . . , vm
n
. . . , vm
}. 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 ≥ m, if not then
there are two distinct vertices va and vb where va ∈ V1 \ S1 and vb ∈ V2 \ S2 such
that 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 | ≥ m.
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
n
Lemma 2.5. For every integer m ≥ 3 and n ≥ 2, a set S = {v1n , v2n , . . . , vm
} is a
strong resolving set of generalized web graph W B0 (Cm , n).
Proof. We prove that for every two distinct vertices vks , vlt ∈ V (W B0 (Cm , n)) \ S
with k, l ∈ [1, m] and s, t ∈ [1, n-1], there exists a vertex s ∈ S which strongly
resolves vks and vlt . There are three possible pairs of vertices.
(1) A pair of vertices (vks , vlt ) with k, l ∈ [1, m], k ̸= l and s, t ∈ [1, n-1], s = t.
For every integer k, l ∈ [1, m] without loss of generality 1 ≤ k < l ≤ m,
t
we obtain the shortest vkt − vln path: vkt , vk+1
, . . . , vlt , vlt+1 , . . . , vln−1 , vln , and
t
shortest vlt − vkn path: vlt , vl−1
, . . . , vkt , vkt+1 , . . . , vkn−1 , vkn . So that vlt ∈ I[vkt , vln ]
and vkt ∈ I[vlt , vkn ].
(2) A pair of vertices (vks , vlt ) with k, l ∈ [1, m], k = l and s, t ∈ [1, n-1], s ̸= t.
For every integer s, t ∈ [1, n-1] without loss of generality 1 ≤ s < t ≤ n-1,
we obtain the shortest vkt − vkn path: vks , vks+1 , . . . , vkt , vkt+1 , . . . , vkn−1 , vkn . So
that vks ∈ I[vks , vkn ].
(3) A pair of vertices (vks , vlt ) with k, l ∈ [1, m], k ̸= l and s, t ∈ [1, n-1], s ̸= t.
For every integer k, l ∈ [1, m] and s, t ∈ [1, n-1] without loss of generality
1 ≤ k < l ≤ m and 1 ≤ s < t ≤ n-1, we obtain the shortest vks − vln
s
t
path: vks , vks+1 , . . . , vl−1
, . . . , vlt−1 , vlt , . . . , vln−1 , vln .
, vlt , . . . , vln−1 , vln or vks , vk+1
So that vlt ∈ I[vks , vln ].
From every possible pairs of vertices, there exists a vertex vln ∈ S with l ∈ [1, m]
which strongly resolves vks ,vlt ∈ V (W B0 (Cm , n)) \ S. Thus S is a strong resolving
set of generalized web graph W B0 (Cm , n).
Theorem 2.2. Let W B0 (Cm , n) be the generalized web graph with m ≥ 3 and n ≥
2. Then sdim(W B0 (Cm , n)) = m.
n
} is a strong resolving
Proof. By using Lemma 2.5, we have a set S = {v1n , v2n , . . . , vm
set of W B0 (Cm , n) graph. According to Lemma 2.4, | S | ≥ m so that S =
n
{v1n , v2n , . . . , vm
} is a strong metric basis of W B0 (Cm , n). Hence sdim(W B0 (Cm , n))
= m.
2.4. The Strong Metric Dimension of a Generalized Flower Graph. Miller
et al. [4] defined the generalized flower graph with p petals (or simply, generalized
f lower graph) F L(G, m, n, p) for m ≥ 3, n ≥ 2 is the graph obtained from the
generalized web graph W B(G, m, n) by connecting each of the p pendant vertices to
the apex with an edge. A petal of F L(G, m, n, p) is the subgraph C that contains the
new edge. In particular, we consider for G ∼
= Cm . We assume that the generalized
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T. A. Putri, T. A. Kusmayadi
1
2
flower graph has a vertex set V (F L)={u, v11 , v21 , . . . , vm
, v12 , v22 , . . . , vm
, . . . , v1n−1 , v2n−1 ,
n−1 n n
n
. . . , vm
, v 1 , v 2 , . . . , vm
}. The generalized flower graph F L(Cm , n, p) can be depicted
as in Figure 3.
Figure 3. Generalized flower graph F L(Cm , n, p)
Lemma 2.6. For every integer m = 3 and n ≥ 2, if S is a strong resolving set of
generalized flower graph F L(C3 , n, p) then | S | ≥ 3(n - 1).
Proof. Consider a pair of vertex (vks , vlt ) with s, t ∈ [1,n], s ̸= t, and k, l ∈ [1,3],
satisfying both of the conditions of Lemma 2.1. According to Property 2.1, we
obtain vks ∈ S or vlt ∈ S, therefore S has at least 3(n − 1) vertices. Hence | S | ≥
3(n − 1).
Lemma 2.7. For every integer m = 3 and n ≥ 2, a set S = {v11 , v21 , v31 , v12 , v22 , v32 , . . . ,
v1n−1 , v2n−1 , v3n−1 } is a strong resolving set of generalized flower graph F L(C3 , n, p).
Proof. We prove that for every two distinct vertices u, v ∈ V (F L(C3 , n, p)) \ S,
there exists a vertex s ∈ S which strongly resolves u and v. There are two possible
pairs of vertices.
(1) A pair of vertices (u, vkn ) with k ∈ [1,3].
For every integer k ∈ [1,3], we obtain the shortest vkn − vk1 path: vkn , u, vk1 . So
that u ∈ I[vkn , vk1 ].
(2) A pair of vertices (vkn , vln ) with k, l ∈ [1,3], k ̸= l.
For every integer k, l ∈ [1,3], without loss of generality 1 ≤ k < l ≤ 3, we
obtain the shortest vkn − vln−1 path: vkn , vln , vln−1 . So that vln ∈ I[vkn , vln−1 ].
From every possible pairs of vertices, there exists a vertex vln−1 ∈ S or vk1 ∈ S with
k, l ∈ [1,3] which strongly resolves u, v ∈ V (F L(C3 , n, p)) \ S. Thus S is a strong
resolving set of generalized flower graph F L(C3 , n, p).
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
Lemma 2.8. For every integer m ≥ 4 and n ≥ 2, if S is a strong resolving set of
generalized flower graph F L(Cm , n, p) then | S | ≥ mn - 2.
Proof. Consider a pair of vertex (vks , vlt ) with s, t ∈ [1,n], s ̸= t, and k, l ∈ [1,m], k
̸= l satisfying both of the conditions of Lemma 2.1. According to Property 2.1, we
obtain vks ∈ S or vlt ∈ S, therefore S has at least mn − 2 vertices. Hence | S | ≥
mn − 2.
1
Lemma 2.9. For every integer m ≥ 4 and n ≥ 2, a set S = {v11 , v21 , . . . , vm
, v12 , v22 , . . . ,
n
n
2
} is a strong resolving set of generalized flower graph
, vm−2
, . . . , v1n , v2n , . . . , vm−3
vm
F L(Cm , n, p).
Proof. We prove that for every two distinct vertices u, v ∈ V (F L(Cm , n, p)) \ S,
there exists a vertex s ∈ S which strongly resolves u and v. There are three possible
pairs of vertices.
n
(1) A pair of vertices (u, vm−1
).
n
For every integer a ∈ [1,n] and b ∈ [1,m-2], we obtain the shortest vm−1
− vba
n
n
path: vm−1
, u, vba . So that u ∈ I[vm−1
, vba ].
n
(2) A pair of vertices (u, vm
).
n
− vba
For every integer a ∈ [1,n] and b ∈ [1,m-2], we obtain the shortest vm
n
n
path: vm
, u, vba . So that u ∈ I[vm
, vba ].
n
n
).
, vm
(3) A pair of vertices (vm−1
n
n
n
n
n
We obtain the shortest vm−1 −v1n path: vm−1
, vm
, v1n . So that vm
∈ I[vm−1
, v1n ].
From every possible pairs of vertices, there exists a vertex vba ∈ S with a ∈ [1,n] dan
b ∈ [1,m-2] which strongly resolves u, v ∈ V (F L(Cm , n, p)) \ S. Thus S is a strong
resolving set of generalized flower graph F L(Cm , n, p).
Theorem 2.3. Let F L(Cm , n, p) be the generalized flower graph. Then for any
integer m ≥ 3 and n ≥ 2,
{
3(n − 1), m = 3;
sdim(F L(Cm , n, p)) =
mn − 2, m ≥ 4.
Proof. For m = 3 and n ≥ 2, by Lemma 2.6 and Lemma 2.7, we have sdim(F L(Cm , n, p))
= 3(n − 1). Next, for m ≥ 4 and n ≥ 2, by Lemma 2.8 and Lemma 2.9, we have
sdim(F L(Cm , n, p)) = mn - 2.
3. Conclusion
According to the discussion above it can be concluded that the strong metric
dimension of a lollipop graph Lm,n , a generalized web graph W B0 (Cm , n), and a
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2017
On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
generalized flower graph F L(Cm , n, p) are as stated in Theorem 2.1, Theorem 2.2,
and Theorem 2.3, respectively.
References
ˇ
1. Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. Cangalovi´
c, Minimal Doubly Resolving Sets and The
Strong Metric Dimension of Some Complex Polytope, Applied Mathematics and Computation
218 (2012), 9790–9801.
ˇ
2. Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. Cangalovi´
c, Strong Metric Dimension: A Survey,
Yugoslav Journal of Operation Research 24(2) (2014), 187–198.
3. 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.
4. Miller, M., O. Phanalasy, and J. Ryan, Antimagicness of Some Families of Generalized Graphs,
Australasian Journal of Combinatorics 53 (2012), 179–190.
5. Oelermann, O. and J. Peters-Fransen, The Strong Metric Dimension of Graphs and Digraphs,
Discrete Applied Mathematics 155 (2007), 356–364.
6. Seb¨o, A. and E. Tannier, On Metric Generators of Graph, Mathematics and Operations Research 29(2) (2004), 383–393.
7. Weisstein, Eric W., CRC Concise Encyclopedia of Mathematics CD-ROM, 2nd ed., CRC Press,
Boca Raton, 2003.
8. Yi, E., On Strong Metric Dimension Graphs and Their Complements, Acta Mathematica Sinica
29(8) (2013), 1479–1492.
8
2017
GRAPH, A GENERALIZED WEB GRAPH, AND A
GENERALIZED FLOWER GRAPH
Tiffani Arzaqi Putri and Tri Atmojo Kusmayadi
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Sebelas Maret University
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 to be the collection of all vertices that belong to some
shortest u − v path. A vertex s strongly resolves two vertices u and v 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]. 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 lollipop Lm,n graph, generalized web W B(G, m, n) graph,
and generalized flower F L(G, m, n, p) graph.
Keywords : strong metric dimension, strong resolving, lollipop graph, generalized web
graph, generalized flower graph
1. Introduction
Let G be a simple connected undirected graph G = (V, E), where V is a set of
vertices, and E is a set of edges. The distance between vertices u and v, i.e. the
length of a shortest u-v path is denoted by d(u, v). A vertex set B={x1 , x2 , . . . , xk }
of G is a resolving set of G if every two distinct vertices of G are resolved by some
vertex of B. The metric basis of G is a resolving set with minimal cardinality. The
metric dimension of G, denoted by dim(G) is the cardinality of its metric basis.
The concept of strong metric dimension was introduced by Seb¨o and Tannier [6]
in 2004. Kratica et al. [2] defined for two vertices u and v in a connected graph
G, 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 u,v ∈ V (G) are strongly resolved by some vertex of S. The
strong metric dimension of G denoted by sdim(G) is the minimum cardinality over
all strong resolving sets of G.
Many researchers have investigated the strong metric dimension to some graph
classes. In 2004 Seb¨o and Tannier [6] observed that the strong metric dimension of
complete graph Kn is n - 1, cycle graph Cn is ⌈ n2 ⌉, and tree is L(T ) - 1, where L(T )
1
On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
denotes the number of leaves of tree. In 2013 Yi [8] determined that the strong
metric dimension of G is 1 if and only if G = Pn . Kusmayadi et al. [3] determined
the strong metric dimension of some related wheel graph such as sunflower graph, tfold wheel graph, helm graph, and friendship graph. In this paper, we determine the
strong metric dimension of a lollipop Lm,n graph, a generalized web W B(G, m, n)
graph, and a generalized flower F L(G, m, n, p) graph.
2. Main Result
2.1. 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). Oelermann and PetersFransen [5] 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. [1].
Lemma 2.1. Let u, v ∈ V(G), u ̸= v,
(1) d(w,v) ≤ d(u,v) for each w such that {u,w} ∈ E(G), and
(2) d(u,w) ≤ d(u,v) for each w such that {v,w} ∈ 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 a strong resolving set of graph G, then for every two
vertices u,v ∈ V(G) which satisfy conditions 1 and 2 of Lemma 2.1, obtained u ∈ S
or v ∈ S.
We denote diam(G) as the diameter of graph G, i.e. the maximal distance between
two vertices in G.
Property 2.2. If S ⊂ V(G) is a strong resolving set of graph G, then for every two
vertices u, v ∈ V(G) such that d(u,v) = diam(G), obtained u ∈ S or v ∈ S.
2.2. The Strong Metric Dimension of a Lollipop Graph. Weisstein [7] defined
the lollipop graph Lm,n for m ≥ 3, n ≥ 2 as a graph obtained by joining a complete
2
2017
On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
graph Km to a path graph Pn with a bridge. We assume that the lollipop graph has
a vertex set V (Lm,n )={v1 , v2 , . . . , vm , u1 , u2 , . . . , un }. The lollipop graph Lm,n can
be depicted as in Figure 1.
Figure 1. Lollipop graph Lm,n
Lemma 2.2. For every integer m ≥ 3 and n ≥ 1, if S is a strong resolving set of
lollipop graph Lm,n then | S | ≥ m − 1.
Proof. Consider a pair of vertex (vi , un ) with i ∈ [1,m-1] satisfying condition d(vi , un )
= n+1 = diam(Lm,n ). According to Property 2.2, we obtain vi ∈ S or un ∈ S,
therefore S has at least m − 1 vertices. Hence | S | ≥ m − 1.
Lemma 2.3. For every integer m ≥ 3 and n ≥ 1, a set S = {v1 , v2 , . . . , vm−1 } is a
strong resolving set of lollipop graph Lm,n .
Proof. For every integer i ∈ [1, m-1] satisfying condition d(un , vi ) = n+1 = diam(Lm,n ),
we obtain the shortest un − vi path: un , un−1 , un−2 , . . . , u1 , vm , vi . So that vi strongly
resolves a pair of vertices (vm ,un ). Thus vm ∈ I[un , vi ].
For a pair of vertices (uk , ul ) with k ̸= l, k, l ∈ [1, n-1], without loss of generality k
< l we obtain for i ∈ [1,m-1], the shortest ul − vi path: ul , ul−1 , . . . , uk , uk−1 , . . . , u1 ,
vm , vi . So that vi strongly resolves a pair of vertices (uk , ul ). Thus uk ∈ I[ul , vi ].
Therefore S = {v1 , v2 , . . . , vm−1 } is a strong resolving set of lollipop graph Lm,n .
Theorem 2.1. Let Lm,n be the lollipop graph with m ≥ 3 and n ≥ 1. Then
sdim(Lm,n ) = m − 1.
Proof. By using Lemma 2.3 a set S = {v1 , v2 , . . . , vm−1 } is a strong resolving set of
lollipop graph Lm,n with m ≥ 3 and n ≥ 1. According to Lemma 2.2, | S | ≥ m - 1
so that S is a strong metric basis of lollipop graph Lm,n . Hence sdim(Lm,n ) = m 1.
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
2.3. The Strong Metric Dimension of a Generalized Web Graph. Miller
et al. [4] defined the generalized web graph W B(G, m, n) for m ≥ 3, n ≥ 2 is
the graph obtained from the generalized pyramid P (G, m) by taking p copies of
Pn (n ≥ 2) and merging an end vertex of a different copy of Pn with each vertex
of the furthermost copy of G from the apex. In this paper we obtain the strong
metric dimension of generalized web graph with G ∼
= Cm and without center vertex
denoted by W B0 (Cm , n). We assume that the generalized web graph has a vertex
1
2
n−1 n n
n
set V (W Bo )={v11 , v21 , . . . , vm
, v12 , v22 , . . . , vm
, . . . , v1n−1 , v2n−1 , . . . , vm
, v 1 , v 2 , . . . , vm
}.
The generalized web graph W B0 (Cm , n) can be depicted as in Figure 2.
Figure 2. Generalized web graph W B0 (Cm , n)
Lemma 2.4. For every integer m ≥ 3 and n ≥ 2, if S is a strong resolving set of
generalized web graph W B0 (Cm , n) then | S | ≥ m.
Proof. Let S is a strong resolving set of generalized web graph W B0 (Cm , n). Suppose
that S contains at most m - 1 vertices, then | S | < m. Let V1 , V2 ⊂ V (W B0 (Cm , n)),
1
2
n−1
, v12 , v22 , . . . , vm
, . . . , v1n−1 , v2n−1 , . . . , vm
} and V2 = {v1n , v2n ,
with V1 = {v11 , v21 , . . . , vm
n
. . . , vm
}. 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 ≥ m, if not then
there are two distinct vertices va and vb where va ∈ V1 \ S1 and vb ∈ V2 \ S2 such
that 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 | ≥ m.
4
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
n
Lemma 2.5. For every integer m ≥ 3 and n ≥ 2, a set S = {v1n , v2n , . . . , vm
} is a
strong resolving set of generalized web graph W B0 (Cm , n).
Proof. We prove that for every two distinct vertices vks , vlt ∈ V (W B0 (Cm , n)) \ S
with k, l ∈ [1, m] and s, t ∈ [1, n-1], there exists a vertex s ∈ S which strongly
resolves vks and vlt . There are three possible pairs of vertices.
(1) A pair of vertices (vks , vlt ) with k, l ∈ [1, m], k ̸= l and s, t ∈ [1, n-1], s = t.
For every integer k, l ∈ [1, m] without loss of generality 1 ≤ k < l ≤ m,
t
we obtain the shortest vkt − vln path: vkt , vk+1
, . . . , vlt , vlt+1 , . . . , vln−1 , vln , and
t
shortest vlt − vkn path: vlt , vl−1
, . . . , vkt , vkt+1 , . . . , vkn−1 , vkn . So that vlt ∈ I[vkt , vln ]
and vkt ∈ I[vlt , vkn ].
(2) A pair of vertices (vks , vlt ) with k, l ∈ [1, m], k = l and s, t ∈ [1, n-1], s ̸= t.
For every integer s, t ∈ [1, n-1] without loss of generality 1 ≤ s < t ≤ n-1,
we obtain the shortest vkt − vkn path: vks , vks+1 , . . . , vkt , vkt+1 , . . . , vkn−1 , vkn . So
that vks ∈ I[vks , vkn ].
(3) A pair of vertices (vks , vlt ) with k, l ∈ [1, m], k ̸= l and s, t ∈ [1, n-1], s ̸= t.
For every integer k, l ∈ [1, m] and s, t ∈ [1, n-1] without loss of generality
1 ≤ k < l ≤ m and 1 ≤ s < t ≤ n-1, we obtain the shortest vks − vln
s
t
path: vks , vks+1 , . . . , vl−1
, . . . , vlt−1 , vlt , . . . , vln−1 , vln .
, vlt , . . . , vln−1 , vln or vks , vk+1
So that vlt ∈ I[vks , vln ].
From every possible pairs of vertices, there exists a vertex vln ∈ S with l ∈ [1, m]
which strongly resolves vks ,vlt ∈ V (W B0 (Cm , n)) \ S. Thus S is a strong resolving
set of generalized web graph W B0 (Cm , n).
Theorem 2.2. Let W B0 (Cm , n) be the generalized web graph with m ≥ 3 and n ≥
2. Then sdim(W B0 (Cm , n)) = m.
n
} is a strong resolving
Proof. By using Lemma 2.5, we have a set S = {v1n , v2n , . . . , vm
set of W B0 (Cm , n) graph. According to Lemma 2.4, | S | ≥ m so that S =
n
{v1n , v2n , . . . , vm
} is a strong metric basis of W B0 (Cm , n). Hence sdim(W B0 (Cm , n))
= m.
2.4. The Strong Metric Dimension of a Generalized Flower Graph. Miller
et al. [4] defined the generalized flower graph with p petals (or simply, generalized
f lower graph) F L(G, m, n, p) for m ≥ 3, n ≥ 2 is the graph obtained from the
generalized web graph W B(G, m, n) by connecting each of the p pendant vertices to
the apex with an edge. A petal of F L(G, m, n, p) is the subgraph C that contains the
new edge. In particular, we consider for G ∼
= Cm . We assume that the generalized
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
1
2
flower graph has a vertex set V (F L)={u, v11 , v21 , . . . , vm
, v12 , v22 , . . . , vm
, . . . , v1n−1 , v2n−1 ,
n−1 n n
n
. . . , vm
, v 1 , v 2 , . . . , vm
}. The generalized flower graph F L(Cm , n, p) can be depicted
as in Figure 3.
Figure 3. Generalized flower graph F L(Cm , n, p)
Lemma 2.6. For every integer m = 3 and n ≥ 2, if S is a strong resolving set of
generalized flower graph F L(C3 , n, p) then | S | ≥ 3(n - 1).
Proof. Consider a pair of vertex (vks , vlt ) with s, t ∈ [1,n], s ̸= t, and k, l ∈ [1,3],
satisfying both of the conditions of Lemma 2.1. According to Property 2.1, we
obtain vks ∈ S or vlt ∈ S, therefore S has at least 3(n − 1) vertices. Hence | S | ≥
3(n − 1).
Lemma 2.7. For every integer m = 3 and n ≥ 2, a set S = {v11 , v21 , v31 , v12 , v22 , v32 , . . . ,
v1n−1 , v2n−1 , v3n−1 } is a strong resolving set of generalized flower graph F L(C3 , n, p).
Proof. We prove that for every two distinct vertices u, v ∈ V (F L(C3 , n, p)) \ S,
there exists a vertex s ∈ S which strongly resolves u and v. There are two possible
pairs of vertices.
(1) A pair of vertices (u, vkn ) with k ∈ [1,3].
For every integer k ∈ [1,3], we obtain the shortest vkn − vk1 path: vkn , u, vk1 . So
that u ∈ I[vkn , vk1 ].
(2) A pair of vertices (vkn , vln ) with k, l ∈ [1,3], k ̸= l.
For every integer k, l ∈ [1,3], without loss of generality 1 ≤ k < l ≤ 3, we
obtain the shortest vkn − vln−1 path: vkn , vln , vln−1 . So that vln ∈ I[vkn , vln−1 ].
From every possible pairs of vertices, there exists a vertex vln−1 ∈ S or vk1 ∈ S with
k, l ∈ [1,3] which strongly resolves u, v ∈ V (F L(C3 , n, p)) \ S. Thus S is a strong
resolving set of generalized flower graph F L(C3 , n, p).
6
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
Lemma 2.8. For every integer m ≥ 4 and n ≥ 2, if S is a strong resolving set of
generalized flower graph F L(Cm , n, p) then | S | ≥ mn - 2.
Proof. Consider a pair of vertex (vks , vlt ) with s, t ∈ [1,n], s ̸= t, and k, l ∈ [1,m], k
̸= l satisfying both of the conditions of Lemma 2.1. According to Property 2.1, we
obtain vks ∈ S or vlt ∈ S, therefore S has at least mn − 2 vertices. Hence | S | ≥
mn − 2.
1
Lemma 2.9. For every integer m ≥ 4 and n ≥ 2, a set S = {v11 , v21 , . . . , vm
, v12 , v22 , . . . ,
n
n
2
} is a strong resolving set of generalized flower graph
, vm−2
, . . . , v1n , v2n , . . . , vm−3
vm
F L(Cm , n, p).
Proof. We prove that for every two distinct vertices u, v ∈ V (F L(Cm , n, p)) \ S,
there exists a vertex s ∈ S which strongly resolves u and v. There are three possible
pairs of vertices.
n
(1) A pair of vertices (u, vm−1
).
n
For every integer a ∈ [1,n] and b ∈ [1,m-2], we obtain the shortest vm−1
− vba
n
n
path: vm−1
, u, vba . So that u ∈ I[vm−1
, vba ].
n
(2) A pair of vertices (u, vm
).
n
− vba
For every integer a ∈ [1,n] and b ∈ [1,m-2], we obtain the shortest vm
n
n
path: vm
, u, vba . So that u ∈ I[vm
, vba ].
n
n
).
, vm
(3) A pair of vertices (vm−1
n
n
n
n
n
We obtain the shortest vm−1 −v1n path: vm−1
, vm
, v1n . So that vm
∈ I[vm−1
, v1n ].
From every possible pairs of vertices, there exists a vertex vba ∈ S with a ∈ [1,n] dan
b ∈ [1,m-2] which strongly resolves u, v ∈ V (F L(Cm , n, p)) \ S. Thus S is a strong
resolving set of generalized flower graph F L(Cm , n, p).
Theorem 2.3. Let F L(Cm , n, p) be the generalized flower graph. Then for any
integer m ≥ 3 and n ≥ 2,
{
3(n − 1), m = 3;
sdim(F L(Cm , n, p)) =
mn − 2, m ≥ 4.
Proof. For m = 3 and n ≥ 2, by Lemma 2.6 and Lemma 2.7, we have sdim(F L(Cm , n, p))
= 3(n − 1). Next, for m ≥ 4 and n ≥ 2, by Lemma 2.8 and Lemma 2.9, we have
sdim(F L(Cm , n, p)) = mn - 2.
3. Conclusion
According to the discussion above it can be concluded that the strong metric
dimension of a lollipop graph Lm,n , a generalized web graph W B0 (Cm , n), and a
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On the Strong Metric Dimension of . . .
T. A. Putri, T. A. Kusmayadi
generalized flower graph F L(Cm , n, p) are as stated in Theorem 2.1, Theorem 2.2,
and Theorem 2.3, respectively.
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