Chapter 2
Graph Theory and Metric
Dimension
2.1
Science and Engineering
Multi discipline teams and multi discipline areas are words that now a days seem
to be important in the scientific research. Indeed this is even more important
if we focus on ancittae scientiae, i.e. sciences, which are used as tools in other
sciences such as Mathematics is with respect to Engineering. Mathematicians
are usually looking for new varieties of problems and Engineers are usually
looking for new arrival solutions. The dialogue of both areas is interesting for
both communities and conducive to to betterment of science and society.
Nevertheless dialogue is not always easy and requires sufficient efforts from
both Mathematicians and Engineers. The translation of a problem arising from
Engineering to mathematical language requires a deep knowledge of the discipline in which a problem is contextualized. Setting up of assumptions and
hypothesis of the model is always a compromise between realism and tractability. Finally the result given by the mathematical models must be validated by
using engineering tools for conducting experimental or simulation investigations.
This thesis deals with two such problems of engineering which are useful in
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Graph Theory and Metric Dimension
communication networks. We model the problem using the theory of graphs,
solve it using graph theory concepts and using this concept the design and
develop a routing algorithm for a clustered network is presented.
2.2
Graph Theory Concepts
A graph G = (V, E) is an ordered pair consisting of a nonempty set V = V (G)
of elements called vertices and a set E = E(G) of unordered pairs of vertices
called edges.
Two vertices u, v ∈ V (G) are said to be adjacent if there is an edge uv ∈
E(G) joining them. The edge uv ∈ E(G) is also said to be incident to vertices
u and v. The degree of a vertex v, denoted by deg(v) is the number of vertices
in V (G) adjacent to it. The minimum and maximum degrees of the graph δ
and ∆ respectively, are defined as the minimum and maximum over the degrees
of all the vertices of the graph.
A uv-path is a sequence of distinct vertices u = v0 , v1 , v2 , . . . , vn = v so that
vi−1 is adjacent to vi for all i, 1 ≤ i ≤ n , such a path is said to be of length
n. A uu-path of length n is a cycle, denoted by Cn . Among all the uv-paths,
the one having minimum length is called the shortest uv-path. The Shortest
uv-path is also called a Geodesic. A triangle formed in a graph G is often called
a cycle of length 3. A graph is said to be connected if there is a path between
every pair of vertices. If the graph is connected, the distance between every two
vertices a and b, denoted by d(a, b), is the length of the shortest path joining
a and b, and the diameter of the graph, denoted by diam G, is the maximum
among the distances between all pairs of vertices of the graph.
The eccentricity of a vertex v of a graph G, denoted by e(G), is the maximum
length of shortest path from vertex v to a vertex of G. The minimum eccentricity
among all the vertices of G is called radius of G and is denoted by r(G). The
vertices u and v, which are at a distance equal to the diameter of the graph G
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Graph Theory and Metric Dimension
are called antipodal vertices.
A vertex is joined to itself, forms a loop in G. Two or more edges join with
the same vertices x and y called multiple edges. A graph having no loops or
multiple edges is called a simple graph. The number of vertices in a graph G is
called the order of G and the number of edges in a graph G is called its size.
A vertex of degree one is called a pendant vertex and the edge incident with
the pendant vertex is called the pendant edge. A vertex of degree zero is called
an isolated vertex. A graph in which all the vertices are of equal degree is called
a regular graph. A graph G is called r-regular graph if the degree of its every
vertex is r. A graph having no cycles is called an acyclic graph or a forest. A
tree is a connected acyclic graph. A tree of order n is denoted by Tn .
A graph H is called a subgraph of G if every vertex of H is a vertex in G
and every edge of H is an edge of G. If H is a subgraph of G and contains all
the edges of G that join two vertices in V (H) then H is said to be the induced
subgraph, denoted by hHi. A maximal connected subgraph of G is called its
component. A complete graph is a simple graph with an edge between every
pair of vertices. A complete graph on n vertices is denoted by Kn . A graph is
bipartite if its vertex set can be partitioned into two non-empty subsets V1 and
V2 so that each edge of G has one end vertex in V1 and other in V2 . A complete
bipartite graph is a bipartite graph in which each vertex of V1 of order m is
adjacent to all the vertices of V2 of order n and is denoted by Km,n . A graph
K1,n is called a star.
Two graphs G1 and G2 are said to be isomorphic if there exists a bijection
f between their vertex sets such that two vertices are adjacent in G1 if and only
if their images under f are adjacent in G2 .
If G is any graph and v is a vertex of G, then G − v is a graph obtained from
G by deleting the vertex v and all the edges incident with v in G. Similarly, if
e is an edge of G, then G − e is the graph obtained from G by just deleting the
edge e in G.
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Graph Theory and Metric Dimension
A subset S of the vertex set V of a graph G, is called an independent set if
no two vertices of S are adjacent in G.
Since a graph is a binary relationship within elements on a generic set, it is
not surprising that graphs appear, at least implicitly, in many different contexts
of the scientific knowledge such as Sociology, Economy, or Engineering.
2.3
Metric Dimension
Graph structure can be used to study the various concepts of Navigation in
space. A work place can be denoted as node in the graph, and edges denote the
connections between the places. The problem of minimum machines (or Robots)
to be placed at certain nodes to trace each and every node exactly once is a
classical one. This problem can be solved by using networks where places are
interconnected in which, the 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 the metric dimension and
the set of all minimum possible number of landmarks constitute metric basis.
The machines, where they are placed at nodes of the network, know their
distances to sufficiently large set of landmarks and the positions of these machines on the network are uniquely determined. However there is neither the
concept of direction nor that of visibility. Instead we shall assume that a Robot
navigating on a graph can sense the distance to a set of landmarks. The problem
of finding metric dimension requires a study of metric Geometry.
In metric Geometry [51] and [52], the following concepts are standard. Let
S = {v1 , v2 , . . . , vn } be an ordered set of nodes in metric space M . The S coordinates of a node u is given by the vector consisting of distances
{d(u, v1 ), d(u, v2 ), . . . , d(u, vn )}. The set S is called metric basis for M if,
1. No two nodes of M have the same S co-ordinates, and
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Graph Theory and Metric Dimension
2. There is no smaller set satisfying the property 1.
The metric dimension of the space M is the cardinality of any metric basis.
These entire concepts can be applied at once to simple connected, undirected
finite graphs. We give a formal definition of the notion of metric dimension
introduced by F. Harary and R.A Melter in 1976 [15]. The metric dimension of
a graph G = (V, E) is the cardinality of a minimal subset S of V such that for
each pair of vertices u, v of V there is a vertex w in S such that the length of
the shortest path from w to u is different from the length of the shortest path
from w to v.
The co-ordinate associated with each node is based on the distance from the
node to the landmarks, so as to pick just enough landmarks in such a way that
each node has a unique tuple of co-ordinates.
A co-ordinate system on G is defined as follows. Select a set M of nodes
as the landmarks. For each landmark, the co-ordinate of a node v ∈ V in G
having the elements equal to the cardinality of the set M and ith element of
co-ordinate of v is equal to the length of shortest path from the ith landmark to
the vertex v in G.
For example: Consider the Figure 2.1
Figure 2.1: The graph G
Taking landmarks ’a’ and ’b’ the assigned coordinates are shown in the
Figure 2.2.
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Graph Theory and Metric Dimension
Figure 2.2: The graph G(with a and b as landmarks)
Taking the vertices ’a’ and ’c’, the associated coordinate is shown in Figure 2.3
Figure 2.3: The graph G(with a and c as Landmarks)
Remark 2.1. For the above graph we obtained two different sets namely {a, b}
and {a, c} both of these consist of two elements. Further we observed that, there
exists no singleton set of vertices satisfying the property that the associated
coordinates of the vertices of the graph are all distinct with respect to the set.
Thus the above sets are minimal with this property and such a minimum set
of landmarks constitutes a metric basis. The cardinality of this metric basis is
called metric dimension of the graph. On the other hand:
2.4
Definition
The metric dimension of a graph G, denoted by β(G), is defined as the cardinality of a minimal subset S of V having the property that for each pair of vertices
22
Graph Theory and Metric Dimension
u, v there exists a vertex w in S such that d(u, w) 6= d(v, w). The co-ordinate of
each vertex v of V (G) with respect to each landmark ui belong to S is defined
as usual with ith component of v as d(v, ui ) for each i and is of dimension β(G).
Remark 2.2. As the concept of metric dimension is defined on the basis of
the distances between the vertices we here onwards consider only finite simple
connected graphs without loops.
Remark 2.3. We obtained two distinct element subsets a, b and a, c of V (G) for
the graph of Figure 2.1, both of which constitute a metric basis for G. However
the second set is an internally stable set. Such sets often are more important to
compute the actual metric dimension of our future graphs as it minimizes the
number of landmarks by maximizing the possible assignments.
Example: The metric dimension of the Petersen graph G(5, 2) [53] is 3
and the metric basis consists of the vertices u1, u2, u3 : Preliminaries: The
Figure 2.4: Petersen Graph
following are the results according to F.Harary and Melter in [15].
Theorem 2.4. The metric dimension of a complete graph of order n is n − 1,
for n > 1.
Proof. : Since each vertex is adjacent to the other, the set of n − 2 vertices
cannot constitute a metric basis as it yields the same coordinate (1, 1, 1, . . . 1)
|
{z
}
n−1times
23
Graph Theory and Metric Dimension
for exactly two non landmarks, and hence exactly one of these two vertices
should be in the metric basis along with the above n − 2 landmarks.
We now consider a path P n on n vertices. By choosing one of the end vertices
as a landmark we observe that all the vertices are at distinct distances from this
vertex. Hence one of the end vertices constitutes a metric basis. Further if G
is not a tree then it contains a circuit and hence at least two landmarks are
necessary to constitute a metric basis. Thus we have:
Theorem 2.5. [30] The metric dimension of a graph G is 1 if and only if G is
a path.
Remark 2.6. In view of the above Theorem 2.4 and minimality of a cardinality
of set of landmarks for a metric dimension of a graph, it suffices to show β(G)
≤ 2 whenever β(G) = 2.
Remark 2.7. Theorem 2.3 and Theorem 2.4 together imply that for a connected
graph G of order n,1 ≤ β(G) ≤ n − 1, for n > 1. We now see the complexity in
computation of the metric dimension of a graph G.
Consider the graphs shown in the Figure 2.5 (a) and 2.5 (b).
Figure 2.5: The Graph G1 (a), The Graph G2 (b) by adding an edge ‘ae‘
The metric dimension of G1 is 2. The graph G2 is obtained from G1 by
adding an edge 0 ae0 and β(G2 ) = 2. Thus, the addition of an edge retains the
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Graph Theory and Metric Dimension
metric dimension in this case. However it is not true in general. For example
consider the graph K4 − e. The metric dimension of K4 − e is 2 and that of
K4 is3.
Further by adding an edge, even though the metric dimension is increased
or stable in the above cases, this cannot be true for all the graphs. As a counter
example we observe that for the following graph G3 in Figure 2.6: β(G) = 3,
Figure 2.6: The Graph G3
with the metric basis M = {a, b, f }. β(G + ab) = 2, with the metric basis
M = {f, b}.
A similar set of graphs can be obtained for the addition of vertices also.
Thus, while defining the graphs throughout the thesis we concentrate on either
the increasing or the decreasing side (but not both) of metric dimensions.
2.5
Metric Dimension of Wheels
In 1976 F. Harary and R.A. Melter [15] mentioned that the metric dimension
of the wheel W1,n on n + 1 vertices is 2, i.e.β(W1,n ) = 2. However this result
will not hold good for large values of n. In fact by Theorem 2.3 we have metric
dimension of the complete graph Kn for n > 1 is n − 1, and the wheel W1,3
is isomorphic to K4 thus β(W1,3 ) = β(K4 ) = 3 6= 2(Disproved by Shanmukha
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Graph Theory and Metric Dimension
et.al,[24]). By this simple observation, Shanmukha et.al [24] conclude that the
result obtained by F. Harary and R.A. Melter[15] is valid only when n = 4 or
5.
2.6
Graphs with Metric Dimension Two
Graphs with β(G) = 2 have a richer structure. We list a few important properties of such graphs.
Theorem 2.8. [15] If Cn is a cycle of order n, then β(Cn ) = 2.
Theorem 2.9. [30] (Kuller et.al) has shown that the graphs with β(G)=2 contain neither K5 nor K3,3 as a subgraph. This might lead one to conjecture that
such graphs have to be planar; but (Kuller et.al) has exhibited in a non-planar
graph with metric dimension 2. A graph G with β(G) = 2 cannot have either
K5 or K3,3 .
Remark 2.10. The proof can be extended to show that a graph G with β(G) = k
cannot have K2k +1 , as a subgraph.
Example: if k = 2, then β(G) = 2 cannot have K5 as a subgraph (follows from the above Theorem 2.9) The following theorem captures a few other
properties of graphs with metric dimension 2.
Theorem 2.11. [30] Let G = (V, E) be a graph with metric dimension 2 and
let {a, b} ⊂ V be a metric basis in G. The following are true.
1. There is a unique shortest path P between a and b.
2. The degrees of a and b are at most 3.
3. Every other node on p has degree at most 5.
Theorem 2.12. [61] For any graph G with β(G) = 2, all the vertices of triangle
of G cannot have either the same first or the same second co-ordinate.
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Graph Theory and Metric Dimension
Theorem 2.13. [54] For any graph with β(G) = 2, the metric basis S of G
cannot have a vertex V of a sub graph K4 of G.
Theorem 2.14. [30] Let G = (V, E) be a graph with metric dimension two.
Let D be the diameter of G. Then |V | ≤ D2 + 2.
Theorem 2.15. [15] Let G = G(Pm × Pn ) be a Cartesian product of two paths
Pm and Pn . Then β(Pm × Pn ) = 2.
Theorem 2.16. [30] The bound on |V | can be slightly refined based on some of
the observations made earlier in Theorem 2.11. Thus |V | ≤ (D − 1)2 + 8.
Theorem 2.17. [15] If W1,n is a wheel for n ≥ 3, then β(W1,4 ) = β(W1,5 ) = 2.
β(W1,n ) = 2 was the earlier result which was disproved by Shanmukha et.al
and stated that the above result is valid only for n = 4 or 5.
Theorem 2.18. [28] Let G = G(Pm × Cn ) be a Cartesian product of a path Pm
and a cycle Cn .Thenβ(Pm × Cn ) = 2, if n is odd.
Theorem 2.19. [28] Let G = G(Km ×Cn ) be a Cartesian product of a complete
graph Kn and a cycle Cm . Then



2, if n = 1
β(Km × Cn ) =


2, if n = 2 and m isodd
Theorem 2.20. [28] If β(G × H) = 2, then G or H is a path.
Theorem 2.21. [28] For every graph G and for all n ≥ 3, we have β(G×Cn ) =
2 if and only if G is a path and n is odd.
Theorem 2.22. [28] For every integer n ≥ 4 there is a tree Bn with β(Bn ) = 2.
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