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 verticesV(G) ={v1,v2,...,vn}and the set of edgesE(G)={e1,e2,...,en}. For every pair of distinct verticesu, v ∈V(G), an intervalI[u, v] is defined as the collection of all vertices that belong to some shortestu
-vpath. A vertexs∈V(G) is said to be strongly resolved for verticesu, v ∈V(G) ifv ∈
I[u, s] oru∈I[v, s]. A setS⊂V(G) is strong resolving set ofGif every pair of vertices
uandvofGis strongly resolved by some vertices ofS. The smallest cardinality of strong resolving set is called a strong metric basis. The strong metric dimension ofGis defined as the number of the elements of strong metric basis inGdenoted bysdim(G). In this paper, we determine the strong metric dimension of a wheel graphWn and a generalized Jahangir graphJt,m(fortis 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 Seb¨o 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 Seb¨o and Tannier [10] observed that the strong metric dimension of complete
graph Kn isn - 1, cycle graph Cn is ⌈n
2⌉, 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
2. Strong Metric Dimension
LetG be a connected graph with the set of verticesV(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 intervalI[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 setV(Wn) ={c,v1,v2,...,vn}
dengan n≥3. Vertex cis a center that adjacent with vi for i ∈ {1,2,...,n}.
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 leastn−2 vertices.
Lemma 3.2. For every integer n ≥ 4, if S={v1,v2,...,vn−2}, then S is strong re-solving 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). HenceS={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. Forn = 3
LetS={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 eachu, v ∈V(W3), theres ∈S such thatv ∈I[u, s]
or u ∈ I[v, s] then S is strong resolving set with 3 elements. Next we show
thatW3 does not have strong resolving set with 2 elements. SupposeW3 has
strong resolving set with 2 elements, then there are 2 possibilities in taking
vertices ofS, that is,
(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 havesdim(G) ̸= 1 sinceW3 ̸=Pn. It means the minimum
of strong resolving set of W3 has 3 elements. Thus, sdim(G) = 3.
(2) Case 2. Forn ≥ 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 setS ⊂V(J2,m) contain at least one vertex of setYi ={Xij} withXij =
{vi,vj} fori, 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, i̸=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). HenceS ={v1,v2,...,vm−2}is strong resolving
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| ≥ (t
2 −1)m.
Proof. It is known thatS is a strongly resolves of generalized Jahangir graphJt,m, we assume thatS contains most of (t
2-1)m-1 vertices so|S|<(
t
2-1)m. Without loss of generality, we may assume by |S|= a. Sincea ≤(t
2-1)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 (t
2 −1)m vertex.
Lemma 4.4. Fort= 4 andm≥3, a setS={v⌈n
2⌉,v⌈2
n+n
2 ⌉,v⌈ 3n+2n
2 ⌉, ...,v⌈
mn+(m−1)n
2 ⌉} with |S| ≥ (t
2-1)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⌈n
2⌉,
v⌈2n+n
2 ⌉, v⌈ 3n+2n
2 ⌉, ..., v⌈
mn+(m−1)n
2 ⌉} is a strong resolving set of generalized Jahangir
graph J4,m.
Lemma 4.5. For t ≥ 6 and m ≥ 3 with t is even, a set S = {v⌈n
2⌉, v⌈
n
2⌉+2, v⌈
n
2⌉+3, ..., vn, v⌈2n+n
2 ⌉, v⌈2
n+n
2 ⌉+2, v⌈2
n+n
2 ⌉+3, ..., v2n, v⌈3
n+2n
2 ⌉, v⌈3
n+2n
2 ⌉+2, v⌈3
n+2n
2 ⌉+3, ..., v3n,
v⌈mn+(m
−1)n
2 ⌉, v⌈
mn+(m
−1)n
2 ⌉+2, v⌈
mn+(m
−1)n
2 ⌉+3, ...,vmn}with |S| ≥ (
t
2−1)m is a strong resolving set of generalized Jahangir graphJt,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 thatuj ∈ 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) A pair of vertices (vj,vk) with j, k = 1,2,..., ⌈n
2⌉ −1, ⌈
n
2⌉+ 1, n+ 1, n+ 2, ..., ⌈2n+n
2 ⌉ −1, ⌈ 2n+n
2 ⌉+ 1, 2n+ 1, 2n+ 2, ..., ⌈ 3n+2n
2 ⌉ −1, ⌈ 3n+2n
2 ⌉+ 1, ..., (m−1)n+ 1, (m−1)n+ 2, ..., ⌈mn+(2m−1)n⌉ −1, ⌈mn+(m2−1)n⌉+ 1.
For every integer j ∈ {1,2,...,⌈n
2⌉ −1, ⌈
n
2⌉+ 1, n+ 1, n+ 2, ...,⌈ 2n+n
2 ⌉ −1, ⌈2n+n
2 ⌉+ 1, 2n+ 1, 2n+ 2, ..., ⌈ 3n+2n
2 ⌉ −1, ⌈ 3n+2n
2 ⌉+ 1, ..., (m−1)n+ 1, (m−1)n+ 2, ..., ⌈mn+(m2−1)n⌉ −1,⌈mn+(m2−1)n⌉+ 1}andk ∈ {1,2,...,⌈n
2⌉ −2,
n+ 1, n+ 2, ..., ⌈2n+n
2 ⌉ −2, 2n+ 1, 2n+ 2, ...,⌈ 3n+2n
2 ⌉ −2, ..., (m−1)n+ 1, (m−1)n+ 2, ...,⌈mn+(2m−1)n⌉ −2}, d(vk, vk+⌈n
2⌉+1) =⌈
n
2⌉+ 1, we obtain the shortest path between vk and vk+⌈n
2⌉+1 that is vk, vk+1, vk+2, ..., vk+⌈
n
2⌉+1 so that vj ∈ I[vk, vk+⌈n
2⌉+1].
(4) A pair of vertices (c,vj) with j = 1,2,..., ⌈n
2⌉ −1, ⌈
n
2⌉+ 1, n + 1, n+ 2, ..., ⌈2n+n
2 ⌉ −1, ⌈ 2n+n
2 ⌉+ 1, 2n+ 1, 2n+ 2, ..., ⌈ 3n+2n
2 ⌉ −1, ⌈ 3n+2n
2 ⌉+ 1, ..., (m−1)n+ 1, (m−1)n+ 2, ..., ⌈mn+(2m−1)n⌉ −1, ⌈mn+(m2−1)n⌉+ 1.
For every integer j ∈ {1,2,...,⌈n
2⌉ −1, ⌈
n
2⌉+ 1, n+ 1, n+ 2, ...,⌈ 2n+n
2 ⌉ −1, ⌈2n+n
2 ⌉+ 1, 2n+ 1, 2n + 2, ..., ⌈ 3n+2n
2 ⌉ − 1, ⌈ 3n+2n
2 ⌉+ 1, ..., (m −1)n+ 1, (m − 1)n + 2, ..., ⌈mn+(m2−1)n⌉ − 1, ⌈mn+(m2−1)n⌉ + 1} and i ∈ {⌈n
2⌉, ⌈2n+n
2 ⌉, ⌈ 3n+2n
2 ⌉, ..., ⌈
mn+(m−1)n
2 ⌉}, d(c, vi) = ⌈
n
2⌉+ 1, we obtain the short-est path between cand 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,
sdim(Jt,m) =
3, if t= 2 and m= 3; 2, if t= 2 and m= 4;
m−2, if t= 2 and m≥3; (t
2 −1)m, if t≥4 and m ≥3.
Proof. We determine the strong metric dimension of generalized Jahangir graphJt,m
by dividing into four parts according to the value of t and m.
(1) Ift = 2 andm = 3.
Ift = 2 danm = 3. LetS={v1,v2,v3}. We show that for eachu, v ∈ V(J2,3) \Sthere iss∈S such thatv ∈I[u, s] oru∈I[v, s]. Then we obtain interval I[u, s] as follows.
{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 ofW3 has 3 elements. Thus, sdim(J2,3) = 3. (2) Ift = 2 andm = 4.
Ift = 2 and m = 4. Let S={v1,v2}. We show that for eachu, v ∈ V(J2,3) \
S theres ∈ S such thatv ∈ 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) Ift = 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) Ift ≥4 and m ≥ 5.
According to the Lemma 4.3, 4.4, and 4.5 it is known that for everyt ≥2, t
is even and m ≥ 3, we have sdim(Jt,m) = (t
2-1)m.
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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