An Application of Gd-Metric Spaces and Metric Dimension of Graphs

International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
DOI:10.5121/jgraphoc.2015.7101 1
AN APPLICATION OF Gd-METRIC SPACES AND
METRIC DIMENSION OF GRAPHS
Ms. Manjusha R1
and Dr. Sunny Kuriakose A2
1
Research Scholar, Union Christian College, Aluva, Kerala, India - 683102
2
Dean, Federal Institute of Science and Technology, Angamaly, Kerala, India - 683577
Abstract
The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric +
→
×
× R
X
X
X
Gd : where =
= )
/
( W
v
r
X
( )
{ }
)
(
/
)
,
(
),...,
,
(
),
,
( 2
1 G
V
v
v
v
d
v
v
d
v
v
d k ∈ , denoted d
G and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if d
G
numbers of pendant edges are added to the non-basis vertices.
Keywords
Resolving set, Basis, Metric dimension, Infinite Graphs, d
G -metric.
1. Introduction
Graph theory has been used to study the various concepts of navigation in an arbitrary space. A
work place can be denoted as node in a graph, and edges denote the connections between places.
The problem of minimum machine (or Robots) to be placed at certain nodes to trace each and
every node exactly once is worth investigating. The problem can be explained using networks
where places are interconnected in which, a navigating agent moves from one node to another in
the network. The places or nodes of a network where we place the machines (robots) are called
‘landmarks’. The minimum number of machines required to locate each and every node of the
network is termed as “metric dimension” and the set of all minimum possible number of
landmarks constitute “metric basis”.
A discrete metric like generalized metric [14] is defined on the Cartesian product X
X
X ×
× of a
nonempty set X into +
R is used to expand the concept of metric dimension of the graph. The
definition of a generalized metric space is given in 2.6. In this type of spaces a non-negative real
number is assigned to every triplet of elements. Several other studies relevant to metric spaces are
being extended to G-metric spaces. Different generalizations of the usual notion of a metric
space were proposed by several mathematicians such as G¨ahler [17, 18] (called 2-metric spaces)
and Dhage [15, 16] (called D-metric spaces) have pointed out that the results cited by G¨ahler are
independent, rather than generalizations, of the corresponding results in metric spaces. Moreover,
it was shown that Dhage’s notion of D-metric space is flawed by errors and most of the results
established by him and others are invalid. These facts are determined by Mustafa and Sims [14] to
introduce a new concept in the area, called G-metric space.

2
The concept of metric dimension was introduced by P J Slater in [2] and studied independently
by Harary and Melter in [3]. Applications of this navigation of robots in networks are discussed
in [4] and in chemistry, while applications to problems of pattern recognition and image
processing, some of which involve the use of hierarchical structures are given in [5]. Besides
Kuller et.al. provide a formula and a linear time algorithm for computing the metric dimension of
a tree in [1]. On the other hand Chartrand et.al. in [7] characterize the graph with metric
dimension 1, n -1 and n -2. See also in [8] the tight bound on the metric dimension of unicyclic
graphs. Shanmukha and Sooryanarayana [9,10] compute the parameters for wheels, graphs
constructed by joining wheels with paths, complete graphs etc. In 1960’s a natural definition of
the dimension of a graph stated by Paul Erdos and state some related problems and unsolved
problems in [11]. Some other application including coin weighing problems and combinatorial
search and optimization [12]. The metric dimension of the Cartesian products of graph has been
studied by Peters-Fransen and Oellermann [13].
The metric dimension of various classes of graphs is computed in [3, 4, 5, 9, 10]. In [4, 5] the
results of [3] are corrected and in [9, 10] the results of [5] are refined.
2. Preliminaries
The basic definitions and results required in subsequent section are given in this section.
2.1. Definition
A graph )
,
( E
V
G = is an ordered pair consisting of a nonempty set )
(G
V
V = of elements called
vertices and a set )
(G
E
E = of unordered pair of vertices called edges.
Two vertices )
(
, G
V
v
u ∈ are said to be adjacent if there is an edge )
(G
E
uv ∈ joining them. The
edge )
(G
E
uv ∈ is also said to be incident to vertices v
and
u . The degree of a vertex v , denoted by
)
deg(v is the number of vertices in )
(G
V adjacent to it.
An edge of a graph is said to be a pendant edge if it is incident with only one vertex of the graph.
A uv -path is a sequence of distinct vertices v
v
v
v
u n
o =
= ,...,
, 1 so that 1
−
i
v is adjacent to i
v for all
, 1
i i n
≤ ≤ , such a path is said to be of length n. A uu -path of length n is a cycle denoted by
n
C .
A graph is said to be connected if there is a path between every two vertices. A complete graph is
a simple graph (a graph having no loops and parallel edges) in which each pair of distinct vertices
is joined by an edge.
2.2. Definition
A graph G is infinite if the vertex set )
(G
V is infinite. An infinite graph is locally finite if every
vertex has finite degree. An infinite graph is uniformly locally finite if there exists a positive
integer M such that the degree of each vertex is at most M . For example, the infinite path ∞
P is
both locally finite and uniformly locally finite by taking 2
=
M .
2.3. Definition
If G is a connected graph, the distance )
,
( v
u
d between two vertices V(G)
, ∈
v
u is the length of the
shortest path between them. Let { }
k
w
w
w
W ,...,
, 2
1
= be an ordered set of vertices of G and let v be

3
a vertex of G. The representation )
/
( W
v
r of v respect to W is the k-tuple
( )
)
,
(
),...,
,
(
),
,
( 2
1 k
w
v
d
w
v
d
w
v
d . If distinct vertices of G have distinct representations (co-ordinates)
with respect toW , then W is called a resolving set or location set for G. A resolving set of
minimum cardinality is called a basis for G and this cardinality is called the metric dimension or
location number of G and is denoted by dim(G) or ( )
G
β .
For each landmark, the coordinate of a node ‘ v ’ in G having the elements equal to the cardinality
of the set W and th
i element of coordinate of ‘ v ’ equal to the length of the shortest path from the
th
i landmark to the vertex ‘ v ’ in G .
For example, consider the graph G of figure 1. The set }
,
{ 2
1
1 v
v
W = is not a resolving set of G
Figure 1. Figure 2.
Since )
/
(
)
1
,
1
(
)
/
( 1
4
1
3 W
v
r
W
v
r =
= . Similarly, we can show that a set consisting of two distinct
vertices will not give distinct coordinates for the vertices inG . On the other hand, }
,
,
{ 3
2
1
2 v
v
v
W =
form a resolving set for G in figure 2, since the representation for the vertices in G with respect
to 2
W are ( ) ( ) ( ) ( )
1
,
0
,
1
/
,
1
,
1
,
0
/ 2
2
2
1 =
= W
v
r
W
v
r , ( ) ( )
0
,
1
,
1
/ 2
3 =
W
v
r , ( ) ( )
1
,
1
,
1
/ 2
4 =
W
v
r and it is the
minimum resolving set implying that dim( ) 3
G = .
2.4. Remark
A graph can have more than one resolving set. For example consider the graph in figure 3. Here
we obtained two resolving sets namely {a,b} and {a,c}.
Figure 3. A graph with two resolving sets

4
2.5. Definition
Let X be a nonempty set. A d
G - Metric or generalized metric is a function from X
X
X ×
× into
+
R having the following properties:
X
z
y
x
z
y
x
z
y
x
Gd ∈
=
=
= ,
,
for
if
0
)
,
,
(
y
x
X
y
x
y
x
x
Gd ≠
∈
≤ with
,
all
for
)
,
,
(
0
y
z
X
z
y
x
z
y
x
G
y
x
x
G d
d ≠
∈
≤ with
,
,
all
for
)
,
,
(
)
,
,
(
)
variables
three
the
all
in
symmetry
...(
)
,
,
(
)
,
,
(
)
,
,
( =
=
= x
z
y
G
y
z
x
G
z
y
x
G d
d
d &
)
inequality
(Rectangle
,
,
,
)
,
,
(
)
,
,
(
)
,
,
( X
a
z
y
x
z
y
a
G
a
a
x
G
z
y
x
G d
d
d ∈
∀
+
≤
2.6. Illustration
Let )
,
( d
X be a metric space. Define +
→
×
× R
X
X
X
Gd : by
( )=
z
y
x
Gd ,
, )
,
(
)
,
(
)
,
( x
z
d
z
y
d
y
x
d +
+ is a d
G -metric satisfying the above five conditions.
Conversely if )
,
( d
G
X is a d
G -metric space, it is easy to verify that )
,
( d
G
d
X is a metric space
where ( ) ( )
)
,
,
(
)
,
,
(
2
1
, y
y
x
G
y
x
x
G
y
x
d d
d
Gd
+
=
For,
a) ( ) ( ) 0
)
,
,
(
)
,
,
(
2
1
, ≥
+
= y
y
x
G
y
x
x
G
y
x
d d
d
Gd
by (ii)
b) ( ) ( ) 0
)
,
,
(
)
,
,
(
2
1
, =
+
= x
x
x
G
x
x
x
G
x
x
d d
d
Gd
by (i)
c) ( ) ( ) ( )
x
y
d
x
x
y
G
x
y
y
G
y
y
x
G
y
x
x
G
y
x
d d
d G
d
d
d
d
G ,
))
,
,
(
)
,
,
(
(
2
1
)
,
,
(
)
,
,
(
2
1
, =
+
=
+
= by (iv)
d) ( ) ( )
)
,
,
(
)
,
,
(
2
1
, y
y
x
G
y
x
x
G
y
x
d d
d
Gd
+
=
[ ]
)
,
,
(
)
,
,
(
)
,
,
(
)
,
,
(
2
1
y
y
z
G
y
z
z
G
z
z
x
G
z
x
x
G d
d
d
d +
+
+
≤
( ) ( )
y
z
d
z
x
d d
d G
G ,
, +
≤
Since )
,
,
(
)
,
,
(
)
,
,
(
)
,
,
( x
x
z
G
z
z
y
G
x
x
y
G
y
x
x
G d
d
d
d +
≤
=
Similarly )
,
,
(
)
,
,
(
)
,
,
( y
y
z
G
z
z
x
G
y
y
x
G d
d
d +
≤ by (v)
Now we recall a few results already published in [23]
2.7. Theorem [7]
The metric dimension of graph G is 1 if and only if G is a path.
Figure 4. (black colored vertices shows the metric basis for ∞
P )
2.8. Theorem [7]
If n
K is the complete graph with 1
>
n then 1
)
( −
= n
Kn
β .

5
2.9. Theorem [1]
If n
C is a cycle of length 2
>
n , then 2
)
( =
n
C
β .
2.9. Theorem [20]
If G is an infinite graph with finite metric dimension then it is uniformly locally finite.
The infinite graph ∞
2
P is uniformly locally finite with metric dimension equal to two.
Figure 4.
The converse of the above theorem is not true. That is a uniformly locally finite graph need not
have finite metric dimension. For example the infinite comp is uniformly locally finite but its
metric dimension is infinite.
Figure 5.
3. Main Results
3.1. Theorem
The metric dimension of the graph obtained by adding ‘n’ pendant edges to each of the ‘n’
vertices in the complete graph n
K , 2
>
n is same as that of n
K .
Proof: We have ( ) 1
−
= n
Kn
β . Let { } { }
i
n v
v
v
v
W
,...,
, 2
1
= for some n
i
i ≤
≤
1
, be a basis for n
K .
Since every vertices n
K are adjacent to each other the coordinate of (n-1) vertices i
j
v j ≠
, in W
has (n-1) components at which th
j component takes the value ‘0’ and the other components are
1’s with respect to W . Now the vertex W
vi ∉ is adjacent to the vertices in W , its coordinate
vector also has (n-1) components and that will be (1,1,…,1).
Suppose n
m
m
m ,...,
, 2
1 are the pendant edges added correspondingly to the vertices n
v
v
v ,...,
, 2
1 such
that ( ) n
j
u
v
m j
j
j ≤
≤
= 1
,
, . Let the graph obtained in this way is denoted by ∑
+
=
=
n
j
j
n m
K
K
1
.
We know that the coordinate of j
v is (1,1,…,0(j
th
place),1,…,1). So for some j , 1
)
,
( =
j
j u
v
d and

6
since every vertex ( )
n
K
V
v∈ are adjacent to j
v , 2
)
,
( =
j
u
v
d for all those vertices j
v
v ≠ . Hence
the coordinate of j
u will be (2,2,…,1(j
th
place),2,…,2).
That is, the coordinate of 1
u is (1,2,…,2), 2
u is (2,1,…,2)…, n
u is (2,2,…,1) respectively. Thus
the vertices in the graph K obtained by adding ‘n’ pendant edges to each of the vertices in n
K has
distinct coordinates with respect to W . Therefore W itself is the basis for n
K and hence
( ) 1
−
= n
K
β .
3.2. Illustration
Consider 5
K (Figure 6). Here five pendant edges 5
1
),
,
( ≤
≤
= j
u
v
m j
j
j are added at each of the
vertices 5
4
3
2
1 and
,
,
,
, v
v
v
v
v respectively and shown that ( ) 4
1
5
5
1
5 =
−
=








∑
+
=
=
j
j
m
K
K β
β .
Figure 6.
The following corollary is about infinite graph with constant metric dimension.
3.3. Corollary
The above theorem holds for an infinite graph obtained by adding pendant edges
( ) n
j
u
v
m j
j
j ≤
≤
= 1
,
, successively at each j
u . Thus there exist infinite graphs with finite metric
dimension.
The development of uniformly locally finite (ULF)[19] graphs is based on the adjacency operator
A acting on the space of bounded sequences defined on the vertices. It has several applications
in spectral theory. The following theorem gives a simple result on uniformly locally finite graph.
3.4. Theorem
The infinite graph ∑
+
=
∞
=1
j
j
n m
K
K mentioned in theorem 3.1 is uniformly locally finite graph with
finite metric dimension.
Proof: By theorem 3.1 ( ) 1
)
(
1
−
=
∑
+
=
∞
=
n
m
K
K
j
j
n
β
β where ( ) ∞
≤
≤
= j
u
v
m j
j
j 1
,
, . Since every
vertex is adjacent to each other in n
K , 1
)
( −
= n
v
d for )
( n
K
V
v∈ and the degree of the vertices j
u
which is one of the end vertex in each of the edge added to n
K is 2. Now fix a positive integer
1
−
= n
M where 2
>
n . Then M
v
d ≤
)
( for all K
v ∈ . Thus K is uniformly locally finite.

7
3.5. Theorem
Let G be connected graph with k
G =
)
(
β , { }
k
v
v
v
W ,...,
, 2
1
= be the basis and
( )
{ }.
)
(
/
))
,
(
),...,
,
(
,
,
(
)
/
( 2
1 G
V
v
v
v
d
v
v
d
v
v
d
W
v
r
X k ∈
=
= Define generalized metric or d
G -metric
+
→
×
× R
X
X
X
Gd : by
( ) { }
)
,
(
),
,
(
),
,
(
min
,
,
,
,
z
y
d
z
x
d
y
x
d
z
y
x
G k
R
X
z
y
x
d
r
r
r
r
r
r
r
r
r
r
r
r
⊆
∈
=
Where d is the 2-metric defined from +
→
× R
X
X by ∑ −
=
=
k
i
i
i y
x
y
x
d
1
)
,
(
r
r
.
If m
z
y
x
Gd =
)
,
,
(
r
r
r
then the metric dimension of the super graph G
~
obtained by adjoining at most m
pendant edges to the vertices W
v∉ is same as that of G with respect toW . That is ( ) ( )
G
G β
β =
~
.
Proof: Let { }
k
v
v
v
v
W ...,
,
, 3
2
1
= be the basis forG . Then the coordinate space
{ }
)
(
/
)
,
(
),...,
,
(
),
,
(
)
/
( 2
1 G
V
v
v
v
d
v
v
d
v
v
d
W
v
r k ∈
= . Since k
G =
)
(
β , the coordinate of each vertex in G
contains ‘k’ components and they are distinct.
Let ( ) m
z
y
x
Gd =
,
, . Now we add m pendant edges are added to suitable vertices W
v∉ . Suppose
the first pendant edge 1
e is added at W
v j ∉ and ( )
1
,
1 e
j v
v
e = . The coordinate of j
v is
))
,
(
),...,
,
(
),
,
(
( 2
1 k
j
j
j v
v
d
v
v
d
v
v
d and it is distinct from the coordinate of other vertices in G .
Figure 7.
Thus the coordinate of 1
e
v will be )
1
)
,
(
,...,
1
)
,
(
,
1
)
,
(
( 2
1 +
+
+ k
j
j
j v
v
d
v
v
d
v
v
d with respect to W and is
different from all other coordinates of the vertices in G since ))
,
(
),...,
,
(
),
,
(
( 2
1 k
j
j
j v
v
d
v
v
d
v
v
d is
distinct from ))
,
(
),...,
,
(
),
,
(
( 2
1 k
i
i
i v
v
d
v
v
d
v
v
d , j
i
n
i ≠
= ,
,...,
2
,
1 . Hence k
e
G =
+ )
( 1
β . If the second
pendant edge is added at 1
e
v say ( )
2
1
,
2 e
e v
v
e = , then by the same argument as in the case of 1
e
v , the
coordinate of 2
e
v will be )
1
)
,
(
,...,
1
)
,
(
,
1
)
,
(
( 1
1
1 2
1 +
+
+ k
e
e
e v
v
d
v
v
d
v
v
d and it is distinct from all other
coordinates ))
,
(
),...,
,
(
),
,
(
( 2
1 k
j
j
j v
v
d
v
v
d
v
v
d for 1
,
,...,
2
,
1 e
j
n
j =
= . Then obviously the coordinate of
the new vertex is distinct from all other vertices since each component in the coordinate of 2
e
v is
increased by one. Thus k
e
e
G =
+
+ )
( 2
1
β .
Suppose the second pendant edge 2
e is added to j
i
vi ≠
, in 1
e
G + and ( )
2
,
2 e
i v
v
e = . Here also the
coordinate of 2
e
v will be )
1
)
,
(
,...,
1
)
,
(
,
1
)
,
(
( 2
1 +
+
+ k
i
i
i v
v
d
v
v
d
v
v
d . Hence k
e
e
G =
+
+ )
( 2
1
β .
Therefore the result is true for 2
,
1
=
m . Assume that k
e
e
e
G m =
+
+
+
+ − )
....
( 1
2
1
β where
( )
l
e
j
l v
v
e ,
= for 1
,...,
2
,
1
, −
=
∉ m
l
W
v j . If m
e is added at any l
e
v then each of the ‘k’ components in
the coordinate of the vertex m
e
v is increased by one and hence it is distinct from other coordinates.
If m
e is added to any vertex v in G not in W and not the end vertex of any of l
e , 1
,...,
2
,
1 −
= m
l ,
then the coordinate of m
e
v will be )
1
)
,
(
,......
1
)
,
(
,
1
)
,
(
( 2
1 +
+
+ k
v
v
d
v
v
d
v
v
d and distinct from all other
coordinates of the vertices in the super graph 1
2
1 .... −
+
+
+
+ m
e
e
e
G . Thus

8
k
e
e
e
e
G m
m =
+
+
+
+
+ − )
....
( 1
2
1
β . Hence the result is true for m . Thus the theorem is true for any
integral value of +
∈R
z
y
x
Gd )
,
,
(
r
r
r
.
3.6 Example
Consider a 5- vertex Kite say H (Figure 8). There ( ) ( )( ) ( ) ( )
{ }
2
,
3
,
1
,
2
,
1
,
1
0
,
1
,
1
,
0
=
X and
( ) { } X
z
y
x
z
x
d
z
y
d
y
x
d
Min
z
y
x
Gd ∈
= ,
,
,
)
,
(
),
,
(
),
,
(
,
,
r
∑ −
=
=
=
2
1
)
,
(
where
,
2
i
i
i y
x
y
x
d
Thus the minimum number of pendant edges that added to the Kite is 2. If these edges are added
to those vertices which are not in W namely 4
3 and v
v with ( ) 2
2
1 =
+
+ e
e
H
β .
Figure 8.
3.7. Example
Consider 4
C
( ) 2
4 =
C
β with respect to { }
2
1,v
v
W = (Figure 9). Then ( ) ( ) ( ) ( )
{ }
1
,
2
,
2
,
1
,
0
,
1
,
1
,
0
)
/
( =
= W
v
r
X
Figure 9.
By the definition of +
→
×
× R
X
X
X
Gd : , we have
( ) { } X
z
y
x
z
x
d
z
y
d
y
x
d
Min
z
y
x
Gd ∈
= ,
,
,
)
,
(
),
,
(
),
,
(
,
,
r
∑ −
=
=
=
2
1
)
,
(
where
,
1
i
i
i y
x
y
x
d
So one pendant edge is added to 4
C .Suppose the pendant edge is added at W
v ∈
1 and ( )
1
,
1
1 e
v
v
e = .
Then the coordinate of 1
e
v is (1,2) with respect to W , but that is similar to the coordinate of 3
v
(Figure 10). Therefore ( ) W
e
C respect to
with
2
1
4 ≠
+
β .

9
Figure 10.
Similarly if 1
e is added to W
v ∈
2 , the coordinate of 1
e
v will be (2,1) and that is similar to the
coordinate of 4
v (Figure 11). Thus 1
e must be added to any of 3
v or 4
v . It will give a distinct
representation for the coordinates of the vertices in 1
4 e
C + (Figure 12). That is 1
e must be added
to the vertices not in .
W
Figure 11.
Figure 12.
Note: Since W is not unique, ( ) 2
1
4 =
+ e
C
β with respect to another resolving set { }
2
,
1
v
v
W e
=
and ( ) ( )( ) ( ) ( )
{ }
1
,
3
,
2
,
2
,
0
,
2
1
,
1
,
2
,
0
)
/
( =
W
v
r (Figure 13).
Figure 13.
4. Conclusion
This paper gives a measure that can be used in navigation space where the number of robots
required to navigate a work place kept constant. Extension of navigation space will lead us to

10
infinite graphs and its properties. With the help of d
G -metric and its properties we established
general concepts and results.
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