Artikel Umi Nurhidayah M0113054
THE STRONG METRIC DIMENSION OF A WHEEL GRAPH
AND A GENERALIZED JAHANGIR GRAPH
Umi Nurhidayah and Tri Atmojo Kusmayadi
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Sebelas Maret University
Abstract. Let G be a connected graph with the set of vertices V (G) ={v1 ,v2 ,...,vn } and
the set of edges E(G)={e1 ,e2 ,...,en }. For every pair of distinct vertices u, v ∈ V (G), an
interval I[u, v] is defined as the collection of all vertices that belong to some shortest u v path. A vertex s ∈ V (G) is said to be strongly resolved for vertices u, v ∈ V (G) if v ∈
I[u, s] or u ∈ I[v, s]. A set S ⊂ V (G) is strong resolving set of G if every pair of vertices
u and v of G is strongly resolved by some vertices of S. The smallest cardinality of strong
resolving set is called a strong metric basis. The strong metric dimension of G is defined
as the number of the elements of strong metric basis in G denoted by sdim(G). In this
paper, we determine the strong metric dimension of a wheel graph Wn and a generalized
Jahangir graph Jt,m (for t is even and t ≥ 2).
Keywords : strong metric dimension, strong resolving set, wheel graph, generalized
Jahangir graph
1. Introduction
The strong metric dimension was introduced by Sebö and Tannier [10] in 2004.
Oellermann and Peters-Frensen [8] defined For every pair of distinct vertices u, v ∈
V (G), an 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) is said to be strongly
resolved for vertices u, v ∈ V (G) if v ∈ I[u, s] or u ∈ I[v, s]. Suppose that S is
a subset of V (G), a set S is strong resolving set of G if every pair of vertices u
and v of G is strongly resolved by some vertices of S. The smallest cardinality of
strong resolving set is called a strong metric basis. The strong metric dimension of
G is defined as the number of the elements of strong metric basis in G denoted by
sdim(G).
Some authors have investigated the problem of finding strong metric dimension.
In 2004 Sebö and Tannier [10] 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 ) denotes
the number of leaves of tree. In 2013 Yi [11] determined that the metric dimension
of G is 1 if and only if G = Pn . Kusmayadi et al. [6] 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 the strong metric
dimension of wheel graph Wn and generalized Jahangir graph Jt,m .
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The Strong Metric Dimension of . . .
U. Nurhidayah, T. A. Kusmayadi
2. Strong Metric Dimension
Let G be a connected graph with the set of vertices V (G) ={v1 ,v2 ,...,vn } and the
set of edges E(G)={e1 ,e2 ,...,en }. Oellermann and Peters-Frensen defined for every
pair of distinct vertices u, v ∈ V (G), an 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) is said to be strongly resolved for vertices u, v ∈ V (G) if v ∈ I[u, s] or
u ∈ I[v, s]. Suppose that S is a subset of V (G), a set S is strong resolving set of
G if every pair of vertices u and v of G is strongly resolved by some vertices of S.
The smallest cardinality of strong resolving set is called a strong metric basis. The
strong metric dimension of G is defined as the number of the elements of strong
metric basis in G denoted by sdim(G).
We often make use of the following lemma and properties about strong metric
dimension given by Kratica et al. [5].
Lemma 2.1. Let u,v ∈ V(G), with u ̸= v,
(1) d(a,v) ≤ d(u,v) for each vertex a that adjacent with u,
(2) d(u,b) ≤ d(u,v) for each vertex b that adjacent with v,
Then there does not exist vertex s ∈ V(G) with s ̸= u,v that strongly resolves vertices
u dan v.
Property 2.1. If S ⊂ V(G) is a strongly resolving set of graph G, then for every
two vertices u,v ∈ V(G) which satisfy condition 1 and 2 of Lemma 2.1, obtained u
∈ S or v ∈ S.
Property 2.2. If S ⊂ V(G) is a strongly 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.
3. The Strong Metric Dimension of a Wheel Graph
Sudha and Chandra [9] defined the wheel graph, denoted by Wn , is a graph
obtained by combining each point of the cycle graph Cn with exactly one isolated
vertex called the center. Edge is incident to the center is called the radius. We
assume that the generalized Jahangir graph has a vertex set V (Wn ) = {c,v1 ,v2 ,...,vn }
dengan n ≥ 3. Vertex c is a center that adjacent with vi for i ∈ {1,2,...,n}.
Lemma 3.1. For every integer n ≥ 4, if S is a strong resolving set of Wn , then |S|
≥ n-2.
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Proof. We proof this lemma by contradiction. It is known that S is a strong
resolving set of wheel graph Wn , we assume that S contains most of n-2 vertices,
|S| < n-2. Suppose that V ={v1 ,v2 ,...,vn } \ S. Since |S| ≤ n-2, there are 2 distinct
vertices vx ,vy ∈ V (Wn ) \ S such that for each s ∈ S we have vx ∈
/ I[vy ,S] and vy
∈
/ I[vx ,S]. It is contradiction to S as a strong resolving set. Hence, if S is a strong
resolving set of wheel graph Wn then S must contain at least n − 2 vertices.
Lemma 3.2. For every integer n ≥ 4, if S={v1 ,v2 ,...,vn−2 }, then S is strong resolving set of Wn .
Proof. We prove for every two distinct vertices u, v ∈ V (Wn ) \ S there exists
a vertex s ∈ S which strongly resolves u and v. Let us consider pairs of vertices
(c,vi ) and (vi ,vj ), i,j =1,2,...,n with i ̸= j belong to some shortest paths vi ,c,vj or
vi ,c,vj+1 or vi+1 ,c,vj or vi+1 ,c,vj+1 or vi ,vj so that c,vi ,vj ∈ I[vi ,vj ]. Because all
vertices belong to some shortest paths between vi ,vj then vi strongly resolves (c,vi )
and (vi ,vj ). Hence S={v1 ,v2 ,...,vn−2 } is a strong resolving set of wheel graph Wn .
Theorem 3.1. Let Wn be the wheel graph. Then for n ≥ 3,
sdim(Wn ) =
{
3,
for n = 3;
n − 2, for n ≥ 4.
Proof. We divide the proof into two cases according to the values of n.
(1) Case 1. For n = 3
Let S= {v1 , v2 , v3 }. Proved for each u, v ∈ V (W3 ) there s ∈ S such that v ∈
I[u, s] or u ∈ I[v, s]. Then obtained interval I[u, s] as follows.
I[c,v1 ]={c,v1 }, I[v1 ,v2 ]={v1 ,v2 },
I[c,v2 ]={c,v2 }, I[v1 ,v3 ]={v1 ,v3 },
I[c,v3 ]={c,v3 }, I[v2 ,v3 ]={v2 ,v3 }.
It can be seen that for each u, v ∈ V (W3 ), there s ∈ S such that v ∈ I[u, s]
or u ∈ I[v, s] then S is strong resolving set with 3 elements. Next we show
that W3 does not have strong resolving set with 2 elements. Suppose W3 has
strong resolving set with 2 elements, then there are 2 possibilities in taking
vertices of S, that is,
(a) S = {c, vi |i = 1, 2, 3}.
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(b) S = {vi , vi |i, j = 1, 2, 3} with i ̸= j.
According to (a) there are vi ∈
/ I[vj , s] or vj ∈
/ I[ui , s] and according to (b)
there are c ∈
/ I[vi , s] or vi ∈
/ I[c, s]. Consequently, W3 does not have any
strong resolving set with 2 elements. Based on Yi [11], sdim(G) = 1 if only
if G = Pn and we have sdim(G) ̸= 1 since W3 ̸= Pn . It means the minimum
of strong resolving set of W3 has 3 elements. Thus, sdim(G) = 3.
(2) Case 2. For n ≥ 4.
According to Lemma 1 and 2, we have sdim(Wn )= n-2 for n ≥ 4.
4. The Strong Metric Dimension of a Generalized Jahangir Graph
Mojdeh and Ghameshlou [7] defined the generalized Jahangir graph Jt,m with
m ≥ 3 as a graph on tm + 1 vertices consisting of a cycle Ctm and one additional
vertex which is adjacent to m vertices of Ctm at t distance to each other on Cmn .
We assume that the generalized Jahangir graph has a vertex set V (Jt,m ) = {c, u1 ,
u2 , ..., um , v1 , v2 , ..., vn , vn+1 , vn+2 , ..., v2n , ..., v(m−1)n+1 , v(m−1)n+2 , ..., vmn }.
Lemma 4.1. For t = 2 and m ≥ 5, if S is strong resolving set of generalized
Jahangir graph J2,m then |S| ≥ m-2.
Proof. Consider a pair of vertices (vi ,vj ), i, j = 1,2,...,m-2 with i ̸= j which satisfy
conditions 1 and 2 of Lemma 2.1. According to Property 2.1, we obtain vi ∈ S or
vj ∈ S. A set S ⊂ V (J2,m ) contain at least one vertex of set Yi = {Xij } with Xij =
{vi ,vj } for i, j = 1,2,...,m-2, i ̸= j. The amount of Yi set is m-2, therefore S has at
least m-2 vertices. Hence |S| ≥ m-2.
Lemma 4.2. For t = 2 and m ≥ 5, a set S = {v1 ,v2 ,...,vm−2 } with |S| ≥ m-2 is a
strong resolving set of generalized Jahangir graph J2,m .
Proof. We prove that for every two distinct vertices u, v ∈ V (J2,m ) \ S there exist
a vertex s ∈ S which strongly resolves u and v. For every pair of vertices (c,ui ),
(ui ,uj ), (ui ,vm−1 ), and (ui ,vm ), i, j = 1,2,...,m, i ̸= j, belong to some shortest paths
vi , ui , c, uj , vj or vi , ui , c, uj+1 , vj or vi , ui+1 , c, uj , vj or vi , ui+1 , c, uj+1 , vj or
vi , ui+1 , vi+1 , uj , vj or vi , ui , vj so that c, ui , uj , vm−1 , vm ∈ I[vi , vj ]. Because all
vertices belong to some shortest paths between vi and vj then vi strongly resolves
(c,ui ), (ui ,uj ), (ui ,vm−1 ), and (ui ,vm ). Hence S = {v1 ,v2 ,...,vm−2 } is strong resolving
set generalized Jahangir graph J2,m .
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Lemma 4.3. For t ≥ 4 and m ≥ 3 with t is even, if S is a strong resolving of
generalized Jahangir graph Jt,m then |S| ≥ ( 2t − 1)m.
Proof. It is known that S is a strongly resolves of generalized Jahangir graph Jt,m ,
we assume that S contains most of ( 2t -1)m-1 vertices so |S| < ( 2t -1)m. Without loss
of generality, we may assume by |S| = a. Since a ≤ ( 2t -1)m-1 there are two distinct
vertices vp ,vq ∈ V (Jt,m ) \ S such that for every s ∈ S we obtain vp ∈
/ I[vq ,S] and
vq ∈
/ I[vp ,S]. This contradicts with the assumption that S is a strong resolving set
generalized Jahangir graph Jt,m then S must at least ( 2t − 1)m vertex.
Lemma 4.4. For t = 4 and m ≥ 3, a set S = {v⌈ n2 ⌉ , v⌈ 2n+n ⌉ , v⌈ 3n+2n ⌉ , ..., v⌈ mn+(m−1)n ⌉ }
2
2
2
with |S| ≥ ( 2t -1)m is a strong resolving set of generalized Jahangir graph J4,m .
Proof. u, v ∈ V (J4,m ) \ S We prove for every two distinct vertices vertex u, v ∈
V (J4,m ) \ S strongly resolved by vertex s ∈ S. For every pair of vertices (c,ui ),
(ui ,uj ), (ui ,vk−1 ), (ui ,vk+1 ), (vk+1 ,vl−1 ), (ui ,ul+1 ), and (ui ,ul−1 ), i, j = 1,2,...,m dan
k, l ∈ S belong to some shortest paths vk , vk−1 , ui , c, uj , vl+1 , vl or vk , vk−1 , ui , c,
uj , vl−1 , vl or vk , vk+1 , ui , c, uj , vl+1 , vl or vk , vk+1 , ui , c, uj , vl−1 , vl or vk , vk+1 ,
ui , vl−1 , vl so that c, ui , uj , vk−1 , vk+1 , vl−1 , vl+1 ∈ I[vk , vl ]. Because all vertices
belong to some shortest paths between vk and vl then vk strongly resolves (c,ui ),
(ui ,uj ), (ui ,vk−1 ), (ui ,vk+1 ), (vk+1 ,vl−1 ), (ui ,ul+1 ), dan (ui ,ul−1 ). Hence S = {v⌈ n2 ⌉ ,
v⌈ 2n+n ⌉ , v⌈ 3n+2n ⌉ , ..., v⌈ mn+(m−1)n ⌉ } is a strong resolving set of generalized Jahangir
2
2
2
graph J4,m .
Lemma 4.5. For t ≥ 6 and m ≥ 3 with t is even, a set S = {v⌈ n2 ⌉ , v⌈ n2 ⌉+2 , v⌈ n2 ⌉+3 ,
..., vn , v⌈ 2n+n ⌉ , v⌈ 2n+n ⌉+2 , v⌈ 2n+n ⌉+3 , ..., v2n , v⌈ 3n+2n ⌉ , v⌈ 3n+2n ⌉+2 , v⌈ 3n+2n ⌉+3 , ..., v3n ,
2
2
2
2
2
2
v⌈ mn+(m−1)n ⌉ , v⌈ mn+(m−1)n ⌉+2 , v⌈ mn+(m−1)n ⌉+3 , ..., vmn } with |S| ≥ ( 2t − 1)m is a strong
2
2
2
resolving set of generalized Jahangir graph Jt,m .
Proof. We prove that for every two distinct vertices u, v ∈ V (Jt,m )\S , u ̸= v there
s ∈ S such that u ∈ I[v, s] or v ∈ I[u, s]. In taking of vertices u, v ∈ V (Jt,m ) \ S, u
̸= v there are four possibilities.
(1) A pair of vertices (c,uj ) with j = 1,2,...,m.
For every integer j ∈ {1,2,...,m}, j ̸= k and i ∈ {n,2n,...,mn}, d(c, vi ) = 2
with uj and vi are adjacent, we obtain the shortest path between c and vi
that is c,uj ,vi so that uj ∈ I[c, vi ].
(2) A pair of vertices vertex (uj ,uk ) with j, k = 1,2,...,m
For every integer j, k ∈ {1,2,...,m}, j ̸= k and i ∈ {vn ,v2n ,...,vmn }, d(uj , vi )
= 3 with uj dan vi are not adjacent and uk and vi are adjacent, we obtain
the shortest path between uj dan vi that is uj ,c,uk ,vi so that uk ∈ I[uj , vi ].
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(3) A pair of vertices (vj ,vk ) with j, k = 1,2,..., ⌈ n2 ⌉ − 1, ⌈ n2 ⌉ + 1, n + 1, n + 2,
..., ⌈ 2n+n
⌉ − 1, ⌈ 2n+n
⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 3n+2n
⌉ − 1, ⌈ 3n+2n
⌉ + 1, ...,
2
2
2
2
mn+(m−1)n
mn+(m−1)n
⌉ − 1, ⌈
⌉ + 1.
(m − 1)n + 1, (m − 1)n + 2, ..., ⌈
2
2
n
n
For every integer j ∈ {1,2,..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, n + 1, n + 2, ..., ⌈ 2n+n
⌉ − 1,
2
3n+2n
3n+2n
2n+n
⌈ 2 ⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, ..., (m − 1)n + 1,
⌉ − 1, ⌈ mn+(m−1)n
⌉ + 1} and k ∈ {1,2,..., ⌈ n2 ⌉ − 2,
(m − 1)n + 2, ..., ⌈ mn+(m−1)n
2
2
n + 1, n + 2, ..., ⌈ 2n+n
⌉ − 2, 2n + 1, 2n + 2, ..., ⌈ 3n+2n
⌉ − 2, ..., (m − 1)n + 1,
2
2
mn+(m−1)n
⌉ − 2}, d(vk , vk+⌈ n2 ⌉+1 ) = ⌈ n2 ⌉ + 1, we obtain the
(m − 1)n + 2, ..., ⌈
2
shortest path between vk and vk+⌈ n2 ⌉+1 that is vk , vk+1 , vk+2 , ..., vk+⌈ n2 ⌉+1 so
that vj ∈ I[vk , vk+⌈ n2 ⌉+1 ].
(4) A pair of vertices (c,vj ) with j = 1,2,..., ⌈ n2 ⌉ − 1, ⌈ n2 ⌉ + 1, n + 1, n + 2,
⌉ − 1, ⌈ 2n+n
⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 3n+2n
⌉ − 1, ⌈ 3n+2n
⌉ + 1, ...,
..., ⌈ 2n+n
2
2
2
2
mn+(m−1)n
mn+(m−1)n
(m − 1)n + 1, (m − 1)n + 2, ..., ⌈
⌉ − 1, ⌈
⌉ + 1.
2
2
n
n
⌉ − 1,
For every integer j ∈ {1,2,..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, n + 1, n + 2, ..., ⌈ 2n+n
2
2n+n
3n+2n
3n+2n
⌈ 2 ⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, ..., (m − 1)n +
⌉ − 1, ⌈ mn+(m−1)n
⌉ + 1} and i ∈ {⌈ n2 ⌉,
1, (m − 1)n + 2, ..., ⌈ mn+(m−1)n
2
2
⌈ 2n+n
⌉, ⌈ 3n+2n
⌉, ..., ⌈ mn+(m−1)n
⌉}, d(c, vi ) = ⌈ n2 ⌉ + 1, we obtain the short2
2
2
est path between c and vi that are c,uk ,vj ,vj + 1,...,vi ,vi+1 ,...,vj+n−1 ,uk+1 or
c,uk ,vj ,vj + 1,...,vi ,vi+1 ,...,vj+n−1 ,uk−m+1 so that vj ∈ I[c, vi ].
For every possible pairs of vertices, there exist a vertex s ∈ S which strongly
resolves every two distinct vertices of V (Jt,m )\S. Thus S is strong resolving set of
generalized Jahangir graph Jt,m .
Theorem 4.1. Let Jt,m be the generalized Jahangir graph. Then for any integer t
≥ 2, t is even and m ≥ 3,
3,
if t = 2 and m = 3;
2,
if t = 2 and m = 4;
sdim(Jt,m ) =
m − 2,
if t = 2 and m ≥ 3;
( t − 1)m, if t ≥ 4 and m ≥ 3.
2
Proof. We determine the strong metric dimension of generalized Jahangir graph Jt,m
by dividing into four parts according to the value of t and m.
(1) If t = 2 and m = 3.
If t = 2 dan m = 3. Let S={v1 ,v2 ,v3 }. We show that for each u, v ∈ V (J2,3 )
\ S there is s ∈ S such that v ∈ I[u, s] or u ∈ I[v, s]. Then we obtain interval
I[u, s] as follows.
I[c,v1 ] = {c,u1 ,u2 ,v1 }; I[u2 ,v1 ] = {u2 ,v1 } I[c,v2 ] = {c,u2 ,u3 ,v2 }; I[u2 ,v2 ] =
{u2 ,v2 }; I[c,v3 ] = {c,u1 ,u3 ,v3 }; I[u2 ,v3 ] = {c,u1 ,u2 , u3 ,v1 ,v2 ,v3 }; I[u1 ,v1 ] =
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{u1 ,v1 }; I[u3 ,v1 ] = {c,u1 ,u2 , u3 ,v1 ,v2 ,v3 }; I[u1 ,v2 ] = {c,u1 ,u2 , u3 ,v1 ,v2 ,v3 };
I[u3 ,v2 ] = {u3 ,v2 }; I[u1 ,v3 ] = {u1 ,v3 }; I[u3 ,v3 ] = {u3 ,v3 }.
It can be seen that for each u, v ∈ V (J2,3 ), there is s ∈ S such that v ∈
I[u, s] or u ∈ I[v, s] then S is strong resolving set with 3 elements. Next we
show that J2,3 does not have strong resolving set with 2 elements. Suppose
J2,3 has strong resolving set with 2 elements, then there are 5 possibilities in
taking vertices in S, that is,
(a) S = {c, ui |i = 1, 2, 3}.
(b) S = {c, ui |i = 1, 2, 3}.
(c) S = {ui , uj |i, j = 1, 2, 3} with i ̸= j.
(d) S = {vi , vj |i = 1, 2, 3} with i ̸= j.
(e) S = {ui , vi |i = 1, 2, 3}.
According to (a)-(e) there are ui ∈
/ I[vi , s] or vi ∈
/ I[ui , s]. Consequently,
J2,3 doesn’t have any strong resolving set with 2 elements. It means the
minimum strong resolving set of W3 has 3 elements. Thus, sdim(J2,3 ) = 3.
(2) If t = 2 and m = 4.
If t = 2 and m = 4. Let S={v1 ,v2 }. We show that for each u, v ∈ V (J2,3 ) \
S there s ∈ S such that v ∈ I[u, s] or u ∈ I[v, s]. The obtain interval I[u, s]
as follows. I[c,v1 ] = {c,u1 ,u2 ,v1 }; I[u3 ,v2 ] = {u3 ,v2 }; I[c,v2 ] = {c,u2 ,u3 ,v2 };
I[u4 ,v1 ] = {c,u1 ,u2 ,u3 ,v1 ,v4 }; I[u1 ,v1 ] = {u1 ,v1 }; I[u4 ,v2 ] = {c,u2 ,u3 ,u4 ,v2 ,v3 };
I[u1 ,v2 ] = {c,u1 ,u2 ,u3 ,v1 ,v2 }; I[v3 ,v1 ] = {c,u1 ,u2 ,u3 ,u4 ,v1 ,v2 ,v3 ,v4 }; I[u2 ,v1 ]
= {u2 ,v1 }; I[v3 ,v2 ] = {u3 ,v2 ,v3 }; I[u2 ,v2 ] = {u2 ,v2 }; I[v4 ,v1 ] = {u1 ,v1 ,v4 };
I[u3 ,v1 ] = {c,u1 ,u2 ,u3 ,v1 ,v2 }; I[v4 ,v2 ] = {c,u1 ,u2 ,u3 ,u4 ,v1 ,v2 ,v3 ,v4 }.
It can be seen that for each u, v ∈ V (J2,4 ), there s ∈ S such that v ∈ I[u, s]
or u ∈ I[v, s] then S is strong resolving set with 2 elements. The next we
show that J2,4 does not have strong resolving set with 1 elements. Based on
Yi [11], sdim(G) = 1 if only if G = Pn and we have sdim(G) ̸= 1 since J2,4
̸= Pn . It means the minimum of strong resolving set of J2,4 has 2 elements.
Thus, sdim(J2,4 ) = 2.
(3) If t = 2 and m ≥ 5.
According to the Lemma 4.1 and 4.2, it is known that for every t =2 and m
≥ 5, we have sdim(J2,m ) = m − 2.
(4) If t ≥ 4 and m ≥ 5.
According to the Lemma 4.3, 4.4, and 4.5 it is known that for every t ≥ 2, t
is even and m ≥ 3, we have sdim(Jt,m ) = ( 2t -1)m.
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5. Conclusion
According to the discussion, it can be concluded that the metric dimension of
a wheel graph Wn and a generalized Jahangir graph Jt,m are as stated in Theorem
3.1 and Theorem 4.1, respectively.
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[9] Sudha, M., and A. Chandra Babu, Even-even Gracefulness of Some Families of Graphs, IOSR
Journal of Mathematics 8 (2013),7-11.
[10] Sebö, A., and E. Tannier, On Metric Generators of Graphs, Mathematics of Operation Research 29 (2004), no.2, 383-393.
[11] Yi, E., On Strong Metric Dimension of Graphs and Their Complements, Acta Mathematica
Sinica, English Series 29 (2013), no. 8, 1479-1492.
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AND A GENERALIZED JAHANGIR GRAPH
Umi Nurhidayah and Tri Atmojo Kusmayadi
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Sebelas Maret University
Abstract. Let G be a connected graph with the set of vertices V (G) ={v1 ,v2 ,...,vn } and
the set of edges E(G)={e1 ,e2 ,...,en }. For every pair of distinct vertices u, v ∈ V (G), an
interval I[u, v] is defined as the collection of all vertices that belong to some shortest u v path. A vertex s ∈ V (G) is said to be strongly resolved for vertices u, v ∈ V (G) if v ∈
I[u, s] or u ∈ I[v, s]. A set S ⊂ V (G) is strong resolving set of G if every pair of vertices
u and v of G is strongly resolved by some vertices of S. The smallest cardinality of strong
resolving set is called a strong metric basis. The strong metric dimension of G is defined
as the number of the elements of strong metric basis in G denoted by sdim(G). In this
paper, we determine the strong metric dimension of a wheel graph Wn and a generalized
Jahangir graph Jt,m (for t is even and t ≥ 2).
Keywords : strong metric dimension, strong resolving set, wheel graph, generalized
Jahangir graph
1. Introduction
The strong metric dimension was introduced by Sebö and Tannier [10] in 2004.
Oellermann and Peters-Frensen [8] defined For every pair of distinct vertices u, v ∈
V (G), an 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) is said to be strongly
resolved for vertices u, v ∈ V (G) if v ∈ I[u, s] or u ∈ I[v, s]. Suppose that S is
a subset of V (G), a set S is strong resolving set of G if every pair of vertices u
and v of G is strongly resolved by some vertices of S. The smallest cardinality of
strong resolving set is called a strong metric basis. The strong metric dimension of
G is defined as the number of the elements of strong metric basis in G denoted by
sdim(G).
Some authors have investigated the problem of finding strong metric dimension.
In 2004 Sebö and Tannier [10] 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 ) denotes
the number of leaves of tree. In 2013 Yi [11] determined that the metric dimension
of G is 1 if and only if G = Pn . Kusmayadi et al. [6] 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 the strong metric
dimension of wheel graph Wn and generalized Jahangir graph Jt,m .
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The Strong Metric Dimension of . . .
U. Nurhidayah, T. A. Kusmayadi
2. Strong Metric Dimension
Let G be a connected graph with the set of vertices V (G) ={v1 ,v2 ,...,vn } and the
set of edges E(G)={e1 ,e2 ,...,en }. Oellermann and Peters-Frensen defined for every
pair of distinct vertices u, v ∈ V (G), an 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) is said to be strongly resolved for vertices u, v ∈ V (G) if v ∈ I[u, s] or
u ∈ I[v, s]. Suppose that S is a subset of V (G), a set S is strong resolving set of
G if every pair of vertices u and v of G is strongly resolved by some vertices of S.
The smallest cardinality of strong resolving set is called a strong metric basis. The
strong metric dimension of G is defined as the number of the elements of strong
metric basis in G denoted by sdim(G).
We often make use of the following lemma and properties about strong metric
dimension given by Kratica et al. [5].
Lemma 2.1. Let u,v ∈ V(G), with u ̸= v,
(1) d(a,v) ≤ d(u,v) for each vertex a that adjacent with u,
(2) d(u,b) ≤ d(u,v) for each vertex b that adjacent with v,
Then there does not exist vertex s ∈ V(G) with s ̸= u,v that strongly resolves vertices
u dan v.
Property 2.1. If S ⊂ V(G) is a strongly resolving set of graph G, then for every
two vertices u,v ∈ V(G) which satisfy condition 1 and 2 of Lemma 2.1, obtained u
∈ S or v ∈ S.
Property 2.2. If S ⊂ V(G) is a strongly 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.
3. The Strong Metric Dimension of a Wheel Graph
Sudha and Chandra [9] defined the wheel graph, denoted by Wn , is a graph
obtained by combining each point of the cycle graph Cn with exactly one isolated
vertex called the center. Edge is incident to the center is called the radius. We
assume that the generalized Jahangir graph has a vertex set V (Wn ) = {c,v1 ,v2 ,...,vn }
dengan n ≥ 3. Vertex c is a center that adjacent with vi for i ∈ {1,2,...,n}.
Lemma 3.1. For every integer n ≥ 4, if S is a strong resolving set of Wn , then |S|
≥ n-2.
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U. Nurhidayah, T. A. Kusmayadi
Proof. We proof this lemma by contradiction. It is known that S is a strong
resolving set of wheel graph Wn , we assume that S contains most of n-2 vertices,
|S| < n-2. Suppose that V ={v1 ,v2 ,...,vn } \ S. Since |S| ≤ n-2, there are 2 distinct
vertices vx ,vy ∈ V (Wn ) \ S such that for each s ∈ S we have vx ∈
/ I[vy ,S] and vy
∈
/ I[vx ,S]. It is contradiction to S as a strong resolving set. Hence, if S is a strong
resolving set of wheel graph Wn then S must contain at least n − 2 vertices.
Lemma 3.2. For every integer n ≥ 4, if S={v1 ,v2 ,...,vn−2 }, then S is strong resolving set of Wn .
Proof. We prove for every two distinct vertices u, v ∈ V (Wn ) \ S there exists
a vertex s ∈ S which strongly resolves u and v. Let us consider pairs of vertices
(c,vi ) and (vi ,vj ), i,j =1,2,...,n with i ̸= j belong to some shortest paths vi ,c,vj or
vi ,c,vj+1 or vi+1 ,c,vj or vi+1 ,c,vj+1 or vi ,vj so that c,vi ,vj ∈ I[vi ,vj ]. Because all
vertices belong to some shortest paths between vi ,vj then vi strongly resolves (c,vi )
and (vi ,vj ). Hence S={v1 ,v2 ,...,vn−2 } is a strong resolving set of wheel graph Wn .
Theorem 3.1. Let Wn be the wheel graph. Then for n ≥ 3,
sdim(Wn ) =
{
3,
for n = 3;
n − 2, for n ≥ 4.
Proof. We divide the proof into two cases according to the values of n.
(1) Case 1. For n = 3
Let S= {v1 , v2 , v3 }. Proved for each u, v ∈ V (W3 ) there s ∈ S such that v ∈
I[u, s] or u ∈ I[v, s]. Then obtained interval I[u, s] as follows.
I[c,v1 ]={c,v1 }, I[v1 ,v2 ]={v1 ,v2 },
I[c,v2 ]={c,v2 }, I[v1 ,v3 ]={v1 ,v3 },
I[c,v3 ]={c,v3 }, I[v2 ,v3 ]={v2 ,v3 }.
It can be seen that for each u, v ∈ V (W3 ), there s ∈ S such that v ∈ I[u, s]
or u ∈ I[v, s] then S is strong resolving set with 3 elements. Next we show
that W3 does not have strong resolving set with 2 elements. Suppose W3 has
strong resolving set with 2 elements, then there are 2 possibilities in taking
vertices of S, that is,
(a) S = {c, vi |i = 1, 2, 3}.
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The Strong Metric Dimension of . . .
U. Nurhidayah, T. A. Kusmayadi
(b) S = {vi , vi |i, j = 1, 2, 3} with i ̸= j.
According to (a) there are vi ∈
/ I[vj , s] or vj ∈
/ I[ui , s] and according to (b)
there are c ∈
/ I[vi , s] or vi ∈
/ I[c, s]. Consequently, W3 does not have any
strong resolving set with 2 elements. Based on Yi [11], sdim(G) = 1 if only
if G = Pn and we have sdim(G) ̸= 1 since W3 ̸= Pn . It means the minimum
of strong resolving set of W3 has 3 elements. Thus, sdim(G) = 3.
(2) Case 2. For n ≥ 4.
According to Lemma 1 and 2, we have sdim(Wn )= n-2 for n ≥ 4.
4. The Strong Metric Dimension of a Generalized Jahangir Graph
Mojdeh and Ghameshlou [7] defined the generalized Jahangir graph Jt,m with
m ≥ 3 as a graph on tm + 1 vertices consisting of a cycle Ctm and one additional
vertex which is adjacent to m vertices of Ctm at t distance to each other on Cmn .
We assume that the generalized Jahangir graph has a vertex set V (Jt,m ) = {c, u1 ,
u2 , ..., um , v1 , v2 , ..., vn , vn+1 , vn+2 , ..., v2n , ..., v(m−1)n+1 , v(m−1)n+2 , ..., vmn }.
Lemma 4.1. For t = 2 and m ≥ 5, if S is strong resolving set of generalized
Jahangir graph J2,m then |S| ≥ m-2.
Proof. Consider a pair of vertices (vi ,vj ), i, j = 1,2,...,m-2 with i ̸= j which satisfy
conditions 1 and 2 of Lemma 2.1. According to Property 2.1, we obtain vi ∈ S or
vj ∈ S. A set S ⊂ V (J2,m ) contain at least one vertex of set Yi = {Xij } with Xij =
{vi ,vj } for i, j = 1,2,...,m-2, i ̸= j. The amount of Yi set is m-2, therefore S has at
least m-2 vertices. Hence |S| ≥ m-2.
Lemma 4.2. For t = 2 and m ≥ 5, a set S = {v1 ,v2 ,...,vm−2 } with |S| ≥ m-2 is a
strong resolving set of generalized Jahangir graph J2,m .
Proof. We prove that for every two distinct vertices u, v ∈ V (J2,m ) \ S there exist
a vertex s ∈ S which strongly resolves u and v. For every pair of vertices (c,ui ),
(ui ,uj ), (ui ,vm−1 ), and (ui ,vm ), i, j = 1,2,...,m, i ̸= j, belong to some shortest paths
vi , ui , c, uj , vj or vi , ui , c, uj+1 , vj or vi , ui+1 , c, uj , vj or vi , ui+1 , c, uj+1 , vj or
vi , ui+1 , vi+1 , uj , vj or vi , ui , vj so that c, ui , uj , vm−1 , vm ∈ I[vi , vj ]. Because all
vertices belong to some shortest paths between vi and vj then vi strongly resolves
(c,ui ), (ui ,uj ), (ui ,vm−1 ), and (ui ,vm ). Hence S = {v1 ,v2 ,...,vm−2 } is strong resolving
set generalized Jahangir graph J2,m .
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Lemma 4.3. For t ≥ 4 and m ≥ 3 with t is even, if S is a strong resolving of
generalized Jahangir graph Jt,m then |S| ≥ ( 2t − 1)m.
Proof. It is known that S is a strongly resolves of generalized Jahangir graph Jt,m ,
we assume that S contains most of ( 2t -1)m-1 vertices so |S| < ( 2t -1)m. Without loss
of generality, we may assume by |S| = a. Since a ≤ ( 2t -1)m-1 there are two distinct
vertices vp ,vq ∈ V (Jt,m ) \ S such that for every s ∈ S we obtain vp ∈
/ I[vq ,S] and
vq ∈
/ I[vp ,S]. This contradicts with the assumption that S is a strong resolving set
generalized Jahangir graph Jt,m then S must at least ( 2t − 1)m vertex.
Lemma 4.4. For t = 4 and m ≥ 3, a set S = {v⌈ n2 ⌉ , v⌈ 2n+n ⌉ , v⌈ 3n+2n ⌉ , ..., v⌈ mn+(m−1)n ⌉ }
2
2
2
with |S| ≥ ( 2t -1)m is a strong resolving set of generalized Jahangir graph J4,m .
Proof. u, v ∈ V (J4,m ) \ S We prove for every two distinct vertices vertex u, v ∈
V (J4,m ) \ S strongly resolved by vertex s ∈ S. For every pair of vertices (c,ui ),
(ui ,uj ), (ui ,vk−1 ), (ui ,vk+1 ), (vk+1 ,vl−1 ), (ui ,ul+1 ), and (ui ,ul−1 ), i, j = 1,2,...,m dan
k, l ∈ S belong to some shortest paths vk , vk−1 , ui , c, uj , vl+1 , vl or vk , vk−1 , ui , c,
uj , vl−1 , vl or vk , vk+1 , ui , c, uj , vl+1 , vl or vk , vk+1 , ui , c, uj , vl−1 , vl or vk , vk+1 ,
ui , vl−1 , vl so that c, ui , uj , vk−1 , vk+1 , vl−1 , vl+1 ∈ I[vk , vl ]. Because all vertices
belong to some shortest paths between vk and vl then vk strongly resolves (c,ui ),
(ui ,uj ), (ui ,vk−1 ), (ui ,vk+1 ), (vk+1 ,vl−1 ), (ui ,ul+1 ), dan (ui ,ul−1 ). Hence S = {v⌈ n2 ⌉ ,
v⌈ 2n+n ⌉ , v⌈ 3n+2n ⌉ , ..., v⌈ mn+(m−1)n ⌉ } is a strong resolving set of generalized Jahangir
2
2
2
graph J4,m .
Lemma 4.5. For t ≥ 6 and m ≥ 3 with t is even, a set S = {v⌈ n2 ⌉ , v⌈ n2 ⌉+2 , v⌈ n2 ⌉+3 ,
..., vn , v⌈ 2n+n ⌉ , v⌈ 2n+n ⌉+2 , v⌈ 2n+n ⌉+3 , ..., v2n , v⌈ 3n+2n ⌉ , v⌈ 3n+2n ⌉+2 , v⌈ 3n+2n ⌉+3 , ..., v3n ,
2
2
2
2
2
2
v⌈ mn+(m−1)n ⌉ , v⌈ mn+(m−1)n ⌉+2 , v⌈ mn+(m−1)n ⌉+3 , ..., vmn } with |S| ≥ ( 2t − 1)m is a strong
2
2
2
resolving set of generalized Jahangir graph Jt,m .
Proof. We prove that for every two distinct vertices u, v ∈ V (Jt,m )\S , u ̸= v there
s ∈ S such that u ∈ I[v, s] or v ∈ I[u, s]. In taking of vertices u, v ∈ V (Jt,m ) \ S, u
̸= v there are four possibilities.
(1) A pair of vertices (c,uj ) with j = 1,2,...,m.
For every integer j ∈ {1,2,...,m}, j ̸= k and i ∈ {n,2n,...,mn}, d(c, vi ) = 2
with uj and vi are adjacent, we obtain the shortest path between c and vi
that is c,uj ,vi so that uj ∈ I[c, vi ].
(2) A pair of vertices vertex (uj ,uk ) with j, k = 1,2,...,m
For every integer j, k ∈ {1,2,...,m}, j ̸= k and i ∈ {vn ,v2n ,...,vmn }, d(uj , vi )
= 3 with uj dan vi are not adjacent and uk and vi are adjacent, we obtain
the shortest path between uj dan vi that is uj ,c,uk ,vi so that uk ∈ I[uj , vi ].
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(3) A pair of vertices (vj ,vk ) with j, k = 1,2,..., ⌈ n2 ⌉ − 1, ⌈ n2 ⌉ + 1, n + 1, n + 2,
..., ⌈ 2n+n
⌉ − 1, ⌈ 2n+n
⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 3n+2n
⌉ − 1, ⌈ 3n+2n
⌉ + 1, ...,
2
2
2
2
mn+(m−1)n
mn+(m−1)n
⌉ − 1, ⌈
⌉ + 1.
(m − 1)n + 1, (m − 1)n + 2, ..., ⌈
2
2
n
n
For every integer j ∈ {1,2,..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, n + 1, n + 2, ..., ⌈ 2n+n
⌉ − 1,
2
3n+2n
3n+2n
2n+n
⌈ 2 ⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, ..., (m − 1)n + 1,
⌉ − 1, ⌈ mn+(m−1)n
⌉ + 1} and k ∈ {1,2,..., ⌈ n2 ⌉ − 2,
(m − 1)n + 2, ..., ⌈ mn+(m−1)n
2
2
n + 1, n + 2, ..., ⌈ 2n+n
⌉ − 2, 2n + 1, 2n + 2, ..., ⌈ 3n+2n
⌉ − 2, ..., (m − 1)n + 1,
2
2
mn+(m−1)n
⌉ − 2}, d(vk , vk+⌈ n2 ⌉+1 ) = ⌈ n2 ⌉ + 1, we obtain the
(m − 1)n + 2, ..., ⌈
2
shortest path between vk and vk+⌈ n2 ⌉+1 that is vk , vk+1 , vk+2 , ..., vk+⌈ n2 ⌉+1 so
that vj ∈ I[vk , vk+⌈ n2 ⌉+1 ].
(4) A pair of vertices (c,vj ) with j = 1,2,..., ⌈ n2 ⌉ − 1, ⌈ n2 ⌉ + 1, n + 1, n + 2,
⌉ − 1, ⌈ 2n+n
⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 3n+2n
⌉ − 1, ⌈ 3n+2n
⌉ + 1, ...,
..., ⌈ 2n+n
2
2
2
2
mn+(m−1)n
mn+(m−1)n
(m − 1)n + 1, (m − 1)n + 2, ..., ⌈
⌉ − 1, ⌈
⌉ + 1.
2
2
n
n
⌉ − 1,
For every integer j ∈ {1,2,..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, n + 1, n + 2, ..., ⌈ 2n+n
2
2n+n
3n+2n
3n+2n
⌈ 2 ⌉ + 1, 2n + 1, 2n + 2, ..., ⌈ 2 ⌉ − 1, ⌈ 2 ⌉ + 1, ..., (m − 1)n +
⌉ − 1, ⌈ mn+(m−1)n
⌉ + 1} and i ∈ {⌈ n2 ⌉,
1, (m − 1)n + 2, ..., ⌈ mn+(m−1)n
2
2
⌈ 2n+n
⌉, ⌈ 3n+2n
⌉, ..., ⌈ mn+(m−1)n
⌉}, d(c, vi ) = ⌈ n2 ⌉ + 1, we obtain the short2
2
2
est path between c and vi that are c,uk ,vj ,vj + 1,...,vi ,vi+1 ,...,vj+n−1 ,uk+1 or
c,uk ,vj ,vj + 1,...,vi ,vi+1 ,...,vj+n−1 ,uk−m+1 so that vj ∈ I[c, vi ].
For every possible pairs of vertices, there exist a vertex s ∈ S which strongly
resolves every two distinct vertices of V (Jt,m )\S. Thus S is strong resolving set of
generalized Jahangir graph Jt,m .
Theorem 4.1. Let Jt,m be the generalized Jahangir graph. Then for any integer t
≥ 2, t is even and m ≥ 3,
3,
if t = 2 and m = 3;
2,
if t = 2 and m = 4;
sdim(Jt,m ) =
m − 2,
if t = 2 and m ≥ 3;
( t − 1)m, if t ≥ 4 and m ≥ 3.
2
Proof. We determine the strong metric dimension of generalized Jahangir graph Jt,m
by dividing into four parts according to the value of t and m.
(1) If t = 2 and m = 3.
If t = 2 dan m = 3. Let S={v1 ,v2 ,v3 }. We show that for each u, v ∈ V (J2,3 )
\ S there is s ∈ S such that v ∈ I[u, s] or u ∈ I[v, s]. Then we obtain interval
I[u, s] as follows.
I[c,v1 ] = {c,u1 ,u2 ,v1 }; I[u2 ,v1 ] = {u2 ,v1 } I[c,v2 ] = {c,u2 ,u3 ,v2 }; I[u2 ,v2 ] =
{u2 ,v2 }; I[c,v3 ] = {c,u1 ,u3 ,v3 }; I[u2 ,v3 ] = {c,u1 ,u2 , u3 ,v1 ,v2 ,v3 }; I[u1 ,v1 ] =
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{u1 ,v1 }; I[u3 ,v1 ] = {c,u1 ,u2 , u3 ,v1 ,v2 ,v3 }; I[u1 ,v2 ] = {c,u1 ,u2 , u3 ,v1 ,v2 ,v3 };
I[u3 ,v2 ] = {u3 ,v2 }; I[u1 ,v3 ] = {u1 ,v3 }; I[u3 ,v3 ] = {u3 ,v3 }.
It can be seen that for each u, v ∈ V (J2,3 ), there is s ∈ S such that v ∈
I[u, s] or u ∈ I[v, s] then S is strong resolving set with 3 elements. Next we
show that J2,3 does not have strong resolving set with 2 elements. Suppose
J2,3 has strong resolving set with 2 elements, then there are 5 possibilities in
taking vertices in S, that is,
(a) S = {c, ui |i = 1, 2, 3}.
(b) S = {c, ui |i = 1, 2, 3}.
(c) S = {ui , uj |i, j = 1, 2, 3} with i ̸= j.
(d) S = {vi , vj |i = 1, 2, 3} with i ̸= j.
(e) S = {ui , vi |i = 1, 2, 3}.
According to (a)-(e) there are ui ∈
/ I[vi , s] or vi ∈
/ I[ui , s]. Consequently,
J2,3 doesn’t have any strong resolving set with 2 elements. It means the
minimum strong resolving set of W3 has 3 elements. Thus, sdim(J2,3 ) = 3.
(2) If t = 2 and m = 4.
If t = 2 and m = 4. Let S={v1 ,v2 }. We show that for each u, v ∈ V (J2,3 ) \
S there s ∈ S such that v ∈ I[u, s] or u ∈ I[v, s]. The obtain interval I[u, s]
as follows. I[c,v1 ] = {c,u1 ,u2 ,v1 }; I[u3 ,v2 ] = {u3 ,v2 }; I[c,v2 ] = {c,u2 ,u3 ,v2 };
I[u4 ,v1 ] = {c,u1 ,u2 ,u3 ,v1 ,v4 }; I[u1 ,v1 ] = {u1 ,v1 }; I[u4 ,v2 ] = {c,u2 ,u3 ,u4 ,v2 ,v3 };
I[u1 ,v2 ] = {c,u1 ,u2 ,u3 ,v1 ,v2 }; I[v3 ,v1 ] = {c,u1 ,u2 ,u3 ,u4 ,v1 ,v2 ,v3 ,v4 }; I[u2 ,v1 ]
= {u2 ,v1 }; I[v3 ,v2 ] = {u3 ,v2 ,v3 }; I[u2 ,v2 ] = {u2 ,v2 }; I[v4 ,v1 ] = {u1 ,v1 ,v4 };
I[u3 ,v1 ] = {c,u1 ,u2 ,u3 ,v1 ,v2 }; I[v4 ,v2 ] = {c,u1 ,u2 ,u3 ,u4 ,v1 ,v2 ,v3 ,v4 }.
It can be seen that for each u, v ∈ V (J2,4 ), there s ∈ S such that v ∈ I[u, s]
or u ∈ I[v, s] then S is strong resolving set with 2 elements. The next we
show that J2,4 does not have strong resolving set with 1 elements. Based on
Yi [11], sdim(G) = 1 if only if G = Pn and we have sdim(G) ̸= 1 since J2,4
̸= Pn . It means the minimum of strong resolving set of J2,4 has 2 elements.
Thus, sdim(J2,4 ) = 2.
(3) If t = 2 and m ≥ 5.
According to the Lemma 4.1 and 4.2, it is known that for every t =2 and m
≥ 5, we have sdim(J2,m ) = m − 2.
(4) If t ≥ 4 and m ≥ 5.
According to the Lemma 4.3, 4.4, and 4.5 it is known that for every t ≥ 2, t
is even and m ≥ 3, we have sdim(Jt,m ) = ( 2t -1)m.
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5. Conclusion
According to the discussion, it can be concluded that the metric dimension of
a wheel graph Wn and a generalized Jahangir graph Jt,m are as stated in Theorem
3.1 and Theorem 4.1, respectively.
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