THE METRIC DIMENSION OF FRIENDSHIP GRAPH Fn , LOLLIPOP GRAPH L m,n AND PETERSEN GRAPH P n,m
Bulletin of Mathematics
ISSN Printed: 2087-5126; Online: 2355-8202
Vol. 08, No. 02 (2016), pp.117-124 http://jurnal.bull-math.org
THE METRIC DIMENSION OF FRIENDSHIP
GRAPH F n, LOLLIPOP GRAPH L m ,n
AND PETERSEN GRAPH P n ,m
Mulyono and Wulandari
Abstract.Let u and v are vertices in connected graph G, the distance d(u, v)
is the length of the shortest path between the vertices u and v. For an ordered
subset W = {w 1 , w 2 , w 3 , . . . , w k } of vertices in a connected graph G and a vertex v
∈ V (G), a metric representation of v with respect to W is the k−vector r(v|W ) =
(d(v, w 1 ), d(v, w 2 ), . . . , d(v, w k )). The subset W is a resolving set if r(v|W ) forevery two vertices of G have distinct representations. The minimum cardinality of
resolving set for G is called the metric dimension of G and denoted by dim(G). This
paper is devoted to determine the metric dimension of friendship graph F n , lollipop graph L m,n and Petersen graph P n,m for m = 1. f5.1. INTRODUCTION Metric dimension is one of subjects in graph theory. The problem of studying the metric dimension was firstly introduced by Slater in 1975. Harary and Melter [6] proposed the same concept in their paper ’On the Metric Dimension of a Graph’. This paper focused on a concept namely metric representation that is a way to represent vertex location in a graph.
Let u and v are vertices in connected graph G, the distance d(u, v) is the length of the shortest path between the vertices u and v. For an
Received 16-08-2016, Accepted 21-09-2016. 2010 Mathematics Subject Classification : 05C20, 94C12 Key words and Phrases
: resolving set, basis, metric dimension, friendship graph, lollipop graph, Petersen graph
Mulyono & Wulandari – Metric Dimension
1
, w , w , . . . , w ordered subset W = {w
2 3 k } of vertices in a connected graph
G and a vertex v ∈ V (G), the metric representation of v with respect to W is the k−vector r(v|W ) = (d(v, w ), d(v, w ), . . . , d(v, w k )). The following
1
2 definition about metric dimension was proposed by Harary and Melter [6].
Definition 1.1 The subset W is a resolving set if r(v|W ) for every two vertices of G have distinct representations. A resolving set of minimum cardinality for graph G is called a minimum resolving set or a basis for G.
The metric dimension of G, denoted by dim(G), is the number of basis for G .
The concept of metric dimension has proved to be useful in a variety of fields. Chartrand et al [2] applied the resolving set of metric dimension in chemistry to classify the chemical compound. Khuller et al [12] also applied in robotic navigation. Furthermore, Sebo et al [11] applied in combinatorial search and optimization.
Moreover Hindayani [7] has studied the determining metric dimension of K r + mK s graph. The results are dim(K r + mK s ) = m + (r − 2) for m ≥ 2, s = 1, and dim(K r + mK s ) = (s − 1)m + (r − 1) for m, s ≥ 2. Another work related to the metric dimension is proposed by Permana [9] on the determining metric dimension for some trees in specific shape. He obtained dim(C ) = m(n − 1) for m ≥ 1, n ≥ 2, dim(F ) = m(n − 1)
m,n m,n for m, n ≥ 2, and dim(B m,n ) = m(n − 2) for m ≥ 2, n ≥ 3 .
In this paper we consider the metric dimension of friendship graph F n , lollipop graph L and Petersen graph P for m = 1.
m,n n,m
2. PRELIMINARIES , v , v , . . . A graph G consists of a set of objects V (G) = {v
1
2 3 } called
vertices and other set E(G) = {e , e , e , . . . } whose elements are called
1
2
3
edges and graph is usually denoted as G = (V (G), E(G)) [13]. A graph at least has one vertex and perhaps has no edge. The number of vertices in G denoted by |V (G)| is often called the order of G , while the number of edges denoted by |E(G)| is its size [4].
The edge e = (u, v) is denoted to join the vertices u and v. If e = (u, v) is an edge of a graph G, then u and v are adjacent vertices, while u and e are incident, as are v and e [3].
For a connected graph G, we define the distance d(u, v) between two
Mulyono & Wulandari – Metric Dimension
d (u, v) ≥ 0 for all pairs u, v of vertices of G, and d(u, v) = 0 if and only if u = v [3].
A friendship graph F n is a graph that can be constructed by coalescence n copies of the cycle graph C
3 of length 3 with a common vertex. The friendship graph F n is also planar graph with 2n + 1 vertices and 3n edges.
, v , v , , v The vertices set is V (F n ) = {c, v } and the edges set is E(F n ) =
1
2 2 2n {cv , cv , cv , , cv n } ∪ {v v , v v , . . . , v v , . . . , v v } for n ≥ 2 [1].
1
2
3
2
1
2
3 4 2i−1 2i 2n−1 2n The following figure shows the friendship graph in common.
Figure 1: Friendship graph F n A lollipop graph, denoted by L m,n (shown in Figure 2), is a graph which is constructed by appending a complete graph K m , m ≥ 3, to a pendant vertex of path graph P n . The vertices set denoted as V (L m,n ) =
, u , u , . . . , u , v , v , v , . . . , v {u n m } [10].
1
2
3
1
2
3 Figure 2: Lollipop graph L m,n n −1
A Petersen graph, denoted by P for n ≥ 3 and 1 ≤ m ≤ , is a
n,m
2
, u , . . . , u , v , v , . . . , v 3−regular graph with 2n vertices V (P n,m ) = {u n n }
1
2
1
2
and 3n edges E(P ) = {u u , u v , v v }∀i ∈ {1, 2, . . . , n}, where the
n,m i i +1 i i i i +m
subscripts are reduced by modulo n. The following figure shows the Petersen graph P n,
1 .
Mulyono & Wulandari – Metric Dimension
Figure 3: Petersen graph P n,
1
3. RESULTS
3.1 n
The Metric Dimension of Friendship Graph F
We begin by providing a stronger result what we indicated in the preceding section. Theorem 3.1
For all integer n ≥ 2, dim(F n ) = n Proof 3.1
, v , v , . . . , v We choose a subset W = {v }, and we must
1
3 5 2n−1
show that dim(F ) = n for n ≥ 2. By definition 1.1, we got the representa-
n
tions of vertices in graph F n with respect to W are r (c|W ) = (1, 1, 1, . . . , 1, 1) r
(v
1 |W ) = (0, 2, 2, . . . , 2, 2)
r (v |W ) = (1, 2, 2, . . . , 2, 2)
2
r (v |W ) = (2, 0, 2, . . . , 2, 2)
3
r (v |W ) = (2, 1, 2, . . . , 2, 2)
4
r (v |W ) = (2, 2, 0, . . . , 2, 2)
5 .. ..
. = . r
(v 2n−1 |W ) = (2, 2, 2, . . . , 2, 0) r (v |W ) = (2, 2, 2, . . . , 2, 1)
2n From above, the representations of vertices in graph F n are distinct.
This impiles that W is resolving set, but it is not necessarily the lower bound. Thus the upper bound is dim(F n ) ≤ n.
Now, we show that dim(F n ) ≥ n. Let W = {v , v , v , . . . , v }
1
3 5 2n−1
is a resolving set which is |W | = n. Assume that W is another mini-
1
mum resolving set or we can denote |W | < n. If we choose an ordered set
1 W
, v ⊆ W − {v i }, i is odd, so that there are two vertices v i i ∈ F n such
1
- 1
that r(v |W ) = r(v |W ) = (2, 2, 2, . . . , 2, 2). W is not a resolving set, a
i i +1
1 contradiction with assumption. Thus the lower bound is dim(F n ) ≥ n. Mulyono & Wulandari – Metric Dimension Thus, another strong result is showed in the following theorem.
3.2 The Metric Dimension of Lollipop Graph L m,n
, v
, v
3
, . . . , v
m −1
} is a resolving set which is |W | = m−1. Assume that W
1
is another minimum resolving set or we can denote |W
1
| < m − 1. If we choose an ordered set W
1
⊆ {v
1
2
, v
, v
3
, . . . , v
m
} − {v
i
, v
j
}, 1 ≤ i, j ≤ m, i 6= j, so that there are two vertices v i , v j ∈ L m,n such that r(v i |W ) = r(v j |W ) = (1, 1, 1, . . . , 1, 1). W
1
is not a resolving set, a contradiction with assumption. Thus the lower bound is dim(L m,n ) ≥ m − 1.
From the above proving, we conclude that dim(F n ) = n. We continue to the other strong result.
3.3 The Metric Dimension of Petersen Graph P n,m
2
1
Theorem 3.2 For all integer m ≥ 3 and n ≥ 1, dim(L m,n ) = m − 1
3
Proof 3.2 We choose a subset W = {v
1
, v
2
, v
3
, . . . , v m
−1
}, and we must show that dim(L m,n ) = m − 1 for m ≥ 3, n ≥ 1. By definition 1.1, we got the representations of vertices in graph L m,n with respect to W are r (v
1
|W ) = (0, 1, 1, . . . , 1, 1) r (v
2
|W ) = (1, 0, 1, . . . , 1, 1) r (v
|W ) = (1, 1, 0, . . . , 1, 1) .. . = ..
Now, we show that dim(L m,n ) ≥ m−1. Let W = {v
. r (v m
−1
|W ) = (1, 1, 1, . . . , 1, 0) r (v m |W ) = (1, 1, 1, . . . , 1, 1) r (u
1
|W ) = (1, 2, 2, . . . , 2, 2) r (u
2
|W ) = (2, 3, 3, . . . , 3, 3) r (u
3
|W ) = (3, 4, 4, . . . , 4, 4) .. . = ..
. r (u n
−1
|W ) = (n − 1, n, n, . . . , n, n) r (v n |W ) = (n, n + 1, n + 1, . . . , n + 1, n + 1) From above, the representations of vertices in graph L m,n are distinct.
This impiles that W is resolving set, but it is not necessarily the lower bound. Thus the upper bound is dim(L m,n ) ≤ m − 1.
Theorem 3.3 For the Petersen graph P n,m we have
Mulyono & Wulandari – Metric Dimension
(i) dim (P n,m ) = 2 for m = 1, n is odd, n ≥ 3 dim (ii) (P n,m ) = 3 for m = 1, n is even, n ≥ 3
Proof 3.3 Case (i) Showed that dim(P n,m ) = 2 for m = 1, n is odd, n ≥ 3
n +1
We choose a subset W = {u , u }, k = , and we must show that
1
1 k
2
dim (P n,m ) = 2 for m = 1, n is odd, n ≥ 3 . By definition 1.1, we got the representations of vertices in graph P with respect to W are
n,m
r (u |W ) = (0, k − 1)
1
r (u |W ) = (1, k − 2)
2
r (u
3 |W ) = (2, k − 3) .. ..
. = . r
(u k |W ) = (k − 1, k − 1) r (u |W ) = (k − 1, 1)
k +1 .. ..
. = . r
(u n |W ) = (1, k − 1) r (v |W ) = (1, k)
1
r (v |W ) = (2, k − 1)
2
r (v
3 |W ) = (3, k − 2) .. ..
. = . r (v |W ) = (k, 1)
k
r (v k |W ) = (k, 2)
- 1 .. ..
. = . r
(v n |W ) = (2, k) From above, the representations of vertices in graph P n,m are distinct. This impiles that W is resolving set with |W | = 2. Obtained the upper bound is dim(P n,m ) ≤ 2. For the Petersen graph P n,m , there is no resolving set that the cardinality is one. Thus the lower bound is dim(P n,m ) ≥ 2. Obtained that dim(P n,m ) ≤ 2 and dim(P n,m ) ≥ 2 , therefore dim(P n,m ) = 2.
Case (ii) Showed that dim(P n,m ) = 3 for m = 1, n is even, n ≥ 4
n
- 2
We choose a subset W = {u , u k , u n } with k = , and we must show
1
2
that dim(P n,m ) = 3 for m = 1, n is even, n ≥ 4 . By definition 1.1, we got the representations of vertices in graph P n,m with respect to W are
Mulyono & Wulandari – Metric Dimension
r (u |W ) = (0, k − 1, 1)
1
r (u
2 |W ) = (1, k − 2, 2)
r (u |W ) = (2, k − 3, 3)
3 .. ..
. = . r
(u k |W ) = (k − 1, 0, k − 2) r (u |W ) = (k − 2, 1, k − 3)
k +1 .. ..
. = . r
(u n |W ) = (1, n − k, 0) r (v |W ) = (1, k, 2)
1
r (v |W ) = (2, k − 1, 3)
2
r (v |W ) = (3, k − 2, 4)
3 .. ..
. = . r
(v k |W ) = (k, 1, k − 1) r (v |W ) = (k − 1, 2, k − 2)
k +1 .. ..
. = . r (v n |W ) = (2, k − 1, 1) From above, the representations of vertices in graph P n,m are distinct.
This impiles that W is resolving set, but it is not necessarily the lower bound. Thus the upper bound is dim(P ) ≤ 3.
n,m
Now, we show that dim(P ) ≥ 3. Assume W is another minimum
n,m
1
resolving set of P n,m for m = 1, n is even, n ≥ 4 with |W | < 3. If we choose
1
an ordered set W ⊆ W − {v }, so that there are the same representations
1 k
r (u |W ) = r(v |W ) = (1, 2), r(u n |W ) = r(v n |W ) = (2, 1), r(u k |W ) =
2 1 −1
r (v k |W ) = (2, 3), r(u k |W ) = r(v k |W ) = (3, 2). W
1 is not a resolving −1 +1 +2
set, a contradiction with assumption. Thus the lower bound is dim(P n,m ) ≥ 3.
From the above proving, we conclude that dim(P n,m ) = 3. . This completes the proof of of Theorem 3.3
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Mulyono tional P. Limited, Bangalore.: Mathematics Department, Faculty Mathematics and Natural Science, Universitas Negeri Medan. Wulandari E-mail: mulyono mat@yahoo.com : Mathematics Department, Faculty Mathematics and Natural Science, Universitas Negeri Medan.
E-mail: wulandari 0910@yahoo.com