DIGITAL TEXT BOOK

A STUDY OF DISTANCES IN GRAPH
THEORY

PREFACE
In mathematics and computer science, graph theroy is the study of graphs, which
are mathematical structures used to model pairewise relations between objects.A
graph in this context is made up of vertices or nodes or points and edges or arcs
or lines that connect them.A graph may be undirected,meaning that there is no
distinction between the two vertices associated with each edge,or its edges may be
directed from one vertex to another;Graphs are one of the prime objects of study in
discrete mathematices.

Contents
Introduction 1
1 Preliminaries 2
2 DISTANCES IN GRAPH THEORY 6
2.1 DISTANCES ON VERTICES OF AN UNDIRECTED GRAPH . . . 6
2.2 DISTANCES ON VERTICES OF A DIRECTED GRAPH . . . . . . 15
3 SOME DISTANCE DEFINED GRAPHS 23
Conclusion 44
Bibliography 45
i

Introduction
The concept of distance is basic to human experience. In everyday life it actually
means some degree of closeness of two physical objects or ideas. i.e., length, time
interval, gap, rank difference, coolness or remoteness while the term metric is often
used as a standard for measurement.
In Mathematics, there are severel areas which deals with distances such as Graph
Theory, Algebra, Functional Analysis, Probability Theory etc. In this project we
study the notion of distances in graph theory and some graphs related with these
distances.
The dissertation is organized as follows:
Chapter 1 includes preliminaries which contains some basic definitions in graph
theory. Chapter 2 contains distances on the vertices of graphs which is divided into
two sections namely distances on the vertices of undirected graphs (section 1) and
distances on the vertices of directed graphs (section 2). The first section includes
shortest path distance, minimum weighted path distance, graph diameter, steiner
distance and truncated metric.The second section consists of path quasi-metric,
strong metric, oriented diameter and circular metric. Some of the distances satisfy
the conditions of a metric and others do not. Chapter 3 contains distance defined
graphs. There are five such graphs included in this section. They are k-power of a
graph, isometric subgraph of a graph, geodetic graph, interval monotone graph and
median graph.
1

Chapter 1
Preliminaries
Definition 1.0.1. (Graph)
A graph is an ordered triple G = (V (G), E(G), IG), where V (G) is a nonempty set,
E(G) is a set disjoint from V (G), and IG is an ”incidence” map that associates
with each element of E(G), an unordered pair of elements(same or distinct) of
V (G). Elements of V (G) are called the vertices ( or nodes or points) of G, and
elements E(G) are called the edges (or lines )ofG.
Definition 1.0.2. (Finite graph,Infinite graph)
A graph is called finite if both V (G) and E(G) are finite. A graph that is not finite
is called infinite.
Definition 1.0.3. (Order of a finite graph)
The order of a finite graph is the number of its vertices
Definition 1.0.4. (Size of a finite graph)
The size of a finite graph is the number of its edges.
Definition 1.0.5. (Subgraph of a graph,Super graph of a graph)
A graph H is called a subgraph of G if V (H) ⊆ V (G), E(H) ⊆ E(G), and IH is the
restriction of IG to E(H). If H is a subgraph of G, G is called the supergraph of H.
Definition 1.0.6. (Induced subgraph)
2

3
A subgraph H of G is said to be an induced subgraph of G if each edge of G having
its ends in V (H) is also an edge of H.
A subgraph H of G is a spanning subgraph of G, if V (H) = V (G).
The induced subgraph of G with vertex set S ⊆ V (G) is called the subgraph of G
induced by S and is denoted by G[S]. Let E be the subset of E and let S denote the
subset of V consisting of all the end vertices in G of edges in E . Then the graph
(S, E , IG/E ) is the subgraph of G induced by the edge set E of G. It is denoted by
G[E ]
Definition 1.0.7. (Weighted graph / Network)
A graph together with a function which assigns a positive weight to each edge is
called a weighted graph or network.
Definition 1.0.8. (Adjacent edges, vertices)
Two edges are called adjacent, if they incident on a common vertex.
Two vertices are called adjacent if they are connected by an edge.
Definition 1.0.9. (Loops, Multiple edges )
An edge for which the two ends are the same is called a loop at the common vertex.
A set of two or more edges of a graph G is called a set of multiple or parallel edges
if they have the same ends.
Definition 1.0.10. (Walk, Trail, Path, Cycle)
A walk in a graph G is an alternating sequence W : v0e1v1e2v2 . . . envn of vertices
and edges beginning and ending with vertices in which vi−1 and vi are ends of ei .
A walk is called a trail if all the edges appearing in the walk are distinct. It is called
a path if all the vertices are distinct.
A cycle is a closed trail in which the vertices are all distinct.
The length of a walk is the number of edges in it. A walk of length zero consists of
just a single vertex.
Definition 1.0.11. (clique)

4
A clique of G is a complete subgraph of G.
Definition 1.0.12. (Tree)
A tree is a simple connected graph without cycles.
Definition 1.0.13. (Connected graph)
A graph G = (V, E) is called connected, if for any vertices u, v ∈ V there exists a
(u−v) path. i.e., a sequence of edges uw0 = w0w1, w1w2, w1w2, . . . , wn−1wn = wn−1v
from E such that wi = wj for i = j, i, j ∈ {0, 1, 2, . . . , n} and a graph is called m
-connected if there is no set of m − 1 edges whose removal disconnects the graph. A
connected graph is 1-connected.
Definition 1.0.14. (Complete graph)
A complete graph is a graph in which each pair of vertices is connected by an edge.
Definition 1.0.15. (Bipartite graph)
A graph is bipartite if its vertex set can be partitioned into two nonempty subsets
X and Y such that each edge of G has one end in X and the other in Y . The
pair (X, Y ) is called a bipartition of the bipartite graph.The bipartite graph with
bipartition is denoted by G(X, Y ).
Definition 1.0.16. (Simple graph)
A graph is simple if it has no loops and no multiple edges.
Definition 1.0.17. (Directed graph or Digraph)
A directed graph(or digraph) is an ordered triple (V (D), A(D), ID), where V (D) is
a nonempty set called the set of vertices of D; A(D) is a set disjoint from V (D),
called the set of arcs of D; and ID is an incidence map that associates with each arc
of D an ordered pair of vertices of D.
Definition 1.0.18. (Complement of a graph)
Let G be a simple graph. Then the complement Gc
of G is defined by taking V (Gc
) =
V (G) and making two vertices u and v adjacent in Gc
if, and only if, they are
nonadjacent in G.

5
Definition 1.0.19. (Join of two graphs)
The join of two graphs G and H, denoted by G + H, is obtained from their disjoint
union by adding all edges of the form uv where u ∈ V (G) and v ∈ V (H).
Definition 1.0.20. (Nonseparable graph )
A graph G is nonseparable if it is nontrivial, connected, and has no cut vertices.
Definition 1.0.21. (Block of a graph)
A block of a graph is a maximal nonseparable subgraph of G. If G has no cut vertex,
G itself is a block

Chapter 2
DISTANCES IN GRAPH
THEORY
In Graph Theory, we have different types of distances defined on graphs.The follow-
ing are some of them.
2.1 DISTANCES ON VERTICES OF AN UNDI-
RECTED GRAPH
1. SHORTEST PATH DISTANCE / GEODESIC DISTANCE
Shortest path distance is defined for any u, v ∈ V , as the length of a shortest
u − v path in G, i.e., a u − v geodesic and is denoted by dpath.It is also called
Path metric.
Remark:
dpath is a metric.
Proof
• Clearly dpath(u, v) ≥ 0
6

7
•
dpath(u, v) = 0 ⇐⇒ Length of the shortest path is zero.
⇐⇒ u = v
•
dpath(u, v) = Length of the shortest u − v path in G.
= Length of the shortest v − u path in G.
= dpath(u, v)
• dpath(u, v) ≤ dpath(u, w) + dpath(w, v), for any u, v, w ∈ V.
Let u, v, w be any three vertices in V. Consider all of the shortest u − v
paths in G.If w is a vertex on any of these u − v paths, then clearly
dpath(u, v) = dpath(u, w) + dpath(w, v)
Since otherwise this would imply that there is a shortest (u − v) path
which has length lesser than the above path, which is not possible. If w
is not a vertex on this path then we can ﬁnd a shortest u − w path and
a shortest w − v path. Combining these two paths we get a u − v path
which is not a shortest u − v path. Hence in this case
dpath(u, v) ≤ dpath(u, w) + dpath(w, v).
EXAMPLE
x w
wy
u
d
d
d
d
d
d

w w
u
u u
uu
u1 2
7
4 5
6
u3

8
d(u1, u5) = 2, d(u1, u4) = 2, d(u4, u5) = 1
d(u7, u6) = 3, d(u7, u3) = 2, d(u3, u6) = 2
2. MINIMUM WEIGHTED PATH DISTANCE
Minimum weighted path distance on the vertex set of a connected weighted
graph G = (V, E) with positive edge weights (w(e))e∈E is deﬁned as
min
p
e∈P
w(e)
where the minimum is taken over all u − v path P in G and is denoted by
dwpath.It is also called weighted path metric.
Remark
Minimum weighted path distance is a metric.
Proof
•
min
p
e∈P
w(e) ≥ 0
• dwpath(u, v) = 0 ⇐⇒ u = v
For
dwpath(u, v) = 0 ⇐⇒
min
p
e∈P
w(e) = 0
⇐⇒ u = v
• dwpath(u, v) = dwpath(v, u)(Clear from the deﬁnition)

9
• dwpath(u, v) ≤ dwpath(u, w) + dwpath(w, v), for any u, v, w ∈ V
For
Suppose w is not a vertex in any minimum weighted u − v in G. Then
take a minimum weighted u − w path P and a minimum weighted w − v
path P”.
Then
min
p
e∈p
w(e) ≤ min
p
e∈P
w(e) + min
p
e∈P”
w(e)
i.e, dwpath(u, v) ≤ dwpath(u, w) + dwpath(w, v)
Since the union of a minimum weighted u − w path and a minimum
weighted w − v path is a u − v path which need not be a minimum
weighted u − v path.
x w
wy
u
d
d
d
d
d
d

w w
u
u u
uu
u1 2
7
4 5
6
u3
2
1
1
2
1
2 4
2
3
2
2
1
Consider u1 and u5 .
The u1 − u5 path, u1u4u5 is a minimum weighted path of weight 3.
Take u2.
dwpath(u1, u2) = 2, dwpath(u2, u5) = 3, dwpath(u1, u6) = 4
3. GRAPH DIAMETER
Given a connected graph G = (V, E), its graph diameter is the maximal length of

10
shortest u−v paths in G for all u, v ∈ V .i.e, it is the largest value of the path metric
between vertices of G
s
t
u
w u
v
u
u
u
u
u
u
7
1
2
3
45
uu6
G
dpath(u1, u2) = 1, dpath(u1, u3) = 2, dpath(u1, u4) = 2, dpath(u1, u5) = 2
dpath(u3, u6) = 3, dpath(u4, u5) = 1, dpath(u4, u6) = 2
dpath(u4, u7) = 3, dpath(u5, u6) = 1, dpath(u5, u7) = 2, dpath(u6, u7) = 1.
3 is the largest among all the lengths. Therefore the diameter of the above graph is
3.
Remark
A connected graph is vertex-critical if deleting any vertex increases its diameter.
EXAMPLE
Consider the following graph

11

u
w
u
u uv1
v v
v
2
v
3
4
5
G
dpath(v1, v2) = 1, dpath(v1, v3) = 2, dpath(v1, v4) = 2, dpath(v1, v5) = 1, dpath(v2, v3) =
1, dpath(v2, v4) = 2, dpath(v2, v5) = 2, dpath(v3, v4) = 1, dpath(v3, v5) = 2, dpath(v4, v5) =
1
Therefore the diameter is 2.
If we remove any vertex, the diameter will become 3.
Therefore this graph is vertex-critical.
Remark
A connected graph is edge-critical if deleting any edge increases its diameter.
EXAMPLE
Consider the following graph
t t
rstv
v
v
1
3
4
v2
dpath(v1, v2) = 1, dpath(v1, v3) = 2, dpath(v1, v4) = 1
dpath(v2, v3) = 1, dpath(v2, v4) = 2, dpath(v3, v4) = 1.

12
Therefore the graph diameter is 2.
If we delete the edge v1v4, dpath(v1, v4) = 3
diameter of the graph will become 3. similarly if we remove any edge the graph
diameter will become 3.Therefore this graph is edge-critical.
4. STEINER DISTANCE
The Steiner distance of a set S ⊂ V of vertices in a connected graph G = (V, E) is
the minimum size (the number of edges) of a connected subgraph of G containing
S. Those of its vertices which are not in S are called steiner points.
EXAMPLES
d
d
d
d
d
dd
t t
t s
s
v
v
1
3
v
4
G
v
5
v
2
Here V = {v1, v2, v3, v4, v5}. Let S1 = {v1}, S2 = {v1, v2}
S3 = {v1, v2, v3}, S4 = {v1, v3, v5}, S5 = {v3, v5}
Let the steiner distance of a set S be denoted by dsS.Then,ds(S1) = 0, ds(S2) =
1, ds(S3) = 2, ds(S4) = 3, ds(S5) = 2.
G1, G2, G3, G4 and G5 are some connected subgraphs of G having minimum size
containing S1, S2, S3, S4 and S5 respectively.

13
v
G1
v1 s s
G2
V
2V1
s s
t
v v
v
1 2
3
G3

t s
s
v v
vv
1
5 4
3
sr
G
4
s t
st
v v
v
5 4
3
G
5
5. TRUNCATED METRIC
The truncated metric is a metric on the vertex set of a graph,which is equal to 1 for
any two adjacent vertices and is equal to 2 for any non-adjacent vertices. It is the
(1, 2)-B-metric if the degree of any vertex is at most B .
Remark
It is a metric.
For,
Let the truncated metric of two vertices u and v be denoted by dT (u, v)
• Clearly dT (u, v) ≥ 0
• dT (u, v) = dT (v, u)
dT (u, v) =



1 if u and v are adjacent
2 if u and v are non adjacent
=



1 if v and u are adjacent
2 if v and u are non adjacent
= dT (v, u)

14
• dT (u, v) ≤ dT (u, w) + dT (w, v)
dT (u, v) =



1 if u and v are adjacent
2 if u and v are non adjacent
dT (u, w) =



1 if u and w are adjacent
2 if u and w are non adjacent
dT (w, v) =



1 if w and v are adjacent
2 if w and v are non adjacent
Clearly,
dT (u, v) ≤ dT (u, w) + dT (w, v)

u
w
u
u uv1
v v
v
2
v
3
4
5
G
dT (v1, v2) = 1, dT (v1, v3) = 2, dT (v1, v4) = 2
dT (v1, v5) = 1, dT (v2, v3) = 1, dT (v2, v4) = 1
dT (v2, v5) = 2, dT (v3, v4) = 1, dT (v3, v5) = 2, dT (v4, v5) = 1

15
2.2 DISTANCES ON VERTICES OF A DIRECTED
GRAPH
(a) PATH QUASI-METRIC IN DIGRAPHS
Path quasi-metric on the vertex set V of a strongly connected directed
graph D = (V, E) is defined as, for any u, v ∈ V the length of a shortest
directed u − v path in D and is denoted by ddpath.
Remark
Path quasi-metric is a quasi-metric on the vertex set of a strongly con-
nected directed graph D = (V, E).
We have the quasi-metric defined as follows:
Let X be a set. A function d : X × X → R is called a quasi-metrix on
X if d(x, y) ≥ 0 holds ∀ x, y ∈ X with equality occurring if and only if
x = y and d(x, y) ≤ d(x, z) + d(z, y) ∀x, y, z ∈ X.
Remark
ddpath is a quasi-metric.
For
• Clearly, ddpath(x, y) ≥ 0 holds ∀x, y ∈ X.
• Suppose that ddpath(x, y) = 0. Then, x = y (since D is strongly
connected.)
• By the definition of ddpath
ddpath(x, y) ≤ ddpath(x, z) + ddpath(z, y)

16
EXAMPLE
v
u v
v
v
vv1 2
4 v3
D
ddpath(v1, v2) = 1, ddpath(v1, v3) = 2, ddpath(v1, v4) = 1, ddpath(v2, v3) = 1
(b) STRONG METRIC IN DIGRAPHS
The strong metric in digraphs is a metric between vertices u and v of a
strongly connected directed graph D = (V, E), deﬁned as the minimum
size (the number of edges) of a stronly connected subdigraph of D, con-
taining u and v.
Remark
It is a metric.
For,
Let the strong metric between two vertices be denoted by dsm(u, v).
• Clearly dsm(u, v) ≥ 0
• dsm(u, v) = 0 ⇐⇒ u = v.
• dsm(u, v) = minimum size of a strongly connected subdigraph of D

17
containing u and v = minimum size of a strongly connected subdi-
graph of D containing v and u
= dsm(v, u)
• dsm(u, v) = minimum size of a stronly connected subdigraph of D
containing u and v
≤ minimum size of a strongly connected subdigraph of D containing
u and w + minimum size of a strongly connected subdigraph of D
containing w and v
≤ dsm(u, w) + dsm(w, v)
Since by combining a minimum size strongly connected subdigraph
of D containing u and w with a minimum size strongly connected
subdigraph of D containing w and v we get a strongly connected
subdigraph of D containing u and v which need not be of minimum
size.
v
u v
v
v
vv1 2
4 v3
D

18
Consider v1 and v2. The minimum size of the strongly connected subdi-
graph of D containing them is 2. The following is the subdigraph.
u v
v v1 2
D1
(c) ORIENTED DIAMETER
Consider a connected graph G = (V, E) with a strong orientation O.i.e, a
strongly connected directed graph D = (V, E ) with arcs e ∈ E obtained
from edges e ∈ E by orientation O, the oriented diameter of D is the
maximal length of shortest directed (u − v) path in it.
v v
v
v
v
2
1
5
4
3
D
ddpath(v1, v2) = 1, ddpath(v2, v1) = 1, ddpath(v1, v3) = 2, ddpath(v3, v1) =
2, ddpath(v1, v4) = 2, ddpath(v4, v1) = 2, ddpath(v2, v4) = 2, ddpath(v4, v2) =

19
2, ddpath(v3, v4) = 1, ddpath(v4, v3) = 1, ddpath(v3, v5) = 1, ddpath(v5, v3) = 1.
The oriented diameter of D is 2.
Remark
The orientation O is tight if the diameter of G is equal to the oriented
diameter of G.Consider the following directed graph D obtained by giv-
ing the strong orientation O to the graph G.
v
u v
v
v
vv1 2
4 v3
D
ddpath(v1, v2) = 1, ddpath(v2, v1) = 1, ddpath(v1, v3) = 2
The oriented diameter of D is 2

20
v
u v
v
G
v
vv1 2
4 v3
dpath(v2, v3) = 1, dpath(v4, v2) = 2, dpath(v3, v4) = 1
Diameter of G is 2
Here diameter of G is equal to the oriented diameter of G. Hence the
orientation O is tight.
(d) CIRCULAR METRIC IN DIGRAPHS
The circular metric on the vertex set V of a strongly connected directed
graph D = (V, E) is deﬁned by
dc(u, v) = ddpath(u, v) + ddpath(v, u),
where ddpath is the path quasi-metric in digraphs.
Remark
It is a metric.
• dc(u, v) ≥ 0
Clearly dc(u, v) = ddpath(u, v) + ddpath(v, u) ≥ 0

21
• dc(u, v) = dc(v, u)
For,
dc(u, v) = ddpath(u, v) + ddpath(v, u)
= ddpath(v, u) + ddpath(u, v)
= dc(v, u)
• dc(u, v) ≤ dc(u, w) + dc(w, v)
For,
We have, dc(u, w) = ddpath(u, w) + ddpath(w, v)
dc(w, v) = ddpath(w, v) + ddpath(v, w)
ddpath(u, v) + ddpath(v, u) ≤ ddpath(u, w) + ddpath(w, u) + ddpath(w, v) +
ddpath(v, w)
dc(u, v) ≤ dc(u, w) + dc(w, v)
Therefore it is a metric.
EXAMPLE
r
r
r
r
D
v v
v v
1 2
34
dc(v1, v2) = ddpath(v1, v2) + ddpath(v2, v1)
= 1 + 2
= 3
= 2 + 1
= 3

22
= 1 + 2
= 3
= 1 + 2
= 3
= 3 + 3
= 6
= 2 + 1
= 3

Chapter 3
SOME DISTANCE DEFINED
GRAPHS
(a) K-POWER OF A GRAPH
The k power of a graph G = (V, E) is the super graph GK
= (V, E ) of G
with edges between all pairs of vertices having path distance atmost k.
EXAMPLES
r r
rqq
v v
G1 2
v
4
v
3 r r
rqq
v v
G1 2
v
4
v
3

2
23

24
d
d
d
d
d
d
d
d
d
dd
H1
r
r r
s
rr
d
d
d
d
d
d
d
d
d
dd
r
r r
s
rr
H1
2
K6=
Result
If G is a connected graph with atleast two edges,then each edge of G2
belongs to a triangle.
proof
Consider a connected graph G with atleast two edges.Here we have to
ﬁnd G2
. So we draw edges between all the vertices u, v ∈ V such that
dpath(u, v) ≤ 2. Then clearly the vertices of each edge of G is connected
by an edge to the end vertex of the adjacent edge. Then clearly each
edge of G belongs to a triangle.
(b) ISOMETRIC SUBGRAPH
A subgraph H of a graph G = (V, E) is called an isometric subgraph if
the path metric between any two points of H is the same as their path
metric in G.

25
EXAMPLE
d
d
d
d
d
dd
t t
t s
s
v v
v
1 2
3
v
4
G
v
5
dpath(v1, v2) = 1, dpath(v2, v3) = 1, dpath(v2, v4) = 1, dpath(v1, v3) = 2,dpath(v1, v5) =
1, dpath(v4, v5) = 1, dpath(v1, v4) = 2, dpath(v2, v5) = 2
v v
v v
5 4
21 H
s qs
st
dpath(v1, v2) = 1, dpath(v2, v4) = 1 dpath(v1, v5) = 1, dpath(v4, v5) = 1, dpath(v1, v4) =
2, dpath(v2, v5) = 2
Lemma 3.0.1. Let G = (V, E) be a graph and v ∈ V . Suppose that any
two nonadjacent vertices x, y ∈ N(v) have a common neighbor other than
v. Then G − v is an isometric subgraph of G.
Proof. It suﬃces to show that removing v from G does not destroy all
geodesics between any two vertices r, s ∈ V − v. So let P be any r − s
geodesic in G. If P does not contain v then we are done. Otherwise, the
vertices on P just before and after v are some x, y ∈ N(v). We cannot
have xy ∈ E since then there would be a shorter r − s path. Thus, by
assumption, there is some common neighbor w = v of x, y. It follows that

26
P − v + w is an r − s geodesic in G − v which completes the proof.
Deﬁnition 3.0.2. (Distance Preserving Graph)
A connected graph is called distance preserving(dp) if it contains an iso-
metric subgraph of every possible order.
Remark
We can construct, for each possible n and r, an r-regular dp graph with n
nodes. In particular this means that, in order to have vertices of degree r
we must have n ≥ r+1. And if r is odd, then n must be even. Finally, we
must have r ≥ 3 since if r = 2 then a connected graph of this regularity
must be a cycle which is not dp for n ≥ 5. We call the remaining pairs
admissible.
Theorem 3.0.3. For each admissible pair (n, r), there exists a dp graph
with n vertices which is regular of degree r.
Proof. Fix r. We denote the graph we construct on n vertices by Gn.
We will have a special construction for r = 3, so assume for the moment
that r ≥ 4.
For r + 1 ≤ n ≤ 2r, we let Gn = Cr,2r−n + Kn−r where Ks is the
complement of the complete graph on s vertices.
Cn,r can be constructed by using a vertex set V = {1, 2, . . . n} with all
edges ij such that 1 ≥ |i − j| ≥ r/2, together all edges of the form
i(i + n/2) if r is odd. All arithmetic is being done modulo n.
The graph G7 for r = 4 is depicted in the following ﬁgure.

27
t t t
tt

s t

G7
=
It is clear that Gn has n vertices and is regular of degree r. Also above
lemma and the completeness of the underlying bipartite graph show that
removing any vertex from either of the parts Cr,2r−n or Kn−r gives an
isometric subgraph as long as the part from which it is removed has at
least two vertices. Continuing to delete vertices in this fashuion shows
that Gn is dp.
To construct G2r+1 first note that,since n = 2r + 1 is odd,we only
need such a construction for r even. Take the complete bipartite graph
Kr,r and remove r/2 independent edges. Now add a new vertex adjacent
to precisely those vertices where an edge was removed. This vertex will
be called the external vertex, x. For an example, consider the following
graph when r = 4.
s s s
s s s
rs

ss
s
x
G9
=
To construct the isometric subgraphs, first remove one by one all but two
of the vertices adjacent to x, where those two vertices are in different
parts of the underlying bipartite graph Kr,r. These subgraphs will be
isometric by the above lemma and the existence of the two remaining
vertices adjacent to x. Now continue to remove two more vertices not
adjacent to x, one from each part of Kr,r. Since r ≥ 4, there will still be
a vertex of Kr,r not adjacent to x remaining in both parts and so above

28
lemma can be applied. for the set of r + 1 vertices to remove, we take
x ∪ N(x). This leaves the graph Kr/2,r/2 which is isometric. From here,
one continues as in the n ≤ 2r case.
To form Gn for n ≥ 2r + 2, we take advantages of the fact that kr+1
is regular of degree r. So divide n by r + 1 to obtain n = q(r + 1) + t
where q ≥ 2 and 0 ≤ t ≤ r. So
n = (q − 1)(r + 1) + (t + r + 1) = p(r + 1) + s, where p = q − 1 ≥ 1 and
s = t + r + 1 so that r + 1 ≤ s ≤ 2r + 1. Now let Gn = Gs ⊕ K⊕p
r+1 where
the edge removed from Gs is one of the edges of the underlying bipartite
graph. To set up notation,let H0 be the subgraph of Gn obtained from Gs
and H1 . . . Hp be the subgraphs obtained from the copies of Kr+1 listed
so that Hi is adjacent to Hi−1 and Hi+1 for 1 ≤ i r. We also let ux, vy
and wz be edges removed to form H0, H1 and HP respectively, where the
edges added between H0 and H1 are uv and xy.
To construct the isometric subgraphs we remove vertices from H0 in the
same way as did for Gs, taking care never to remove u or x, untill only 4
vertices remain: u, x, and another vertex from each part of the underlying
complete subgraph of Gs. The resulting subgraph will be isometric in Gn
because all geodesics from a vertex of H0 to another Hi go through u or x
which are never removed. Next we continue to remove two more vertices
from Hp, neither of them being w, z. Since r ≥ 4 these subgraphs will
again be isometric since there will always exist a third vertex of Hp present
in the subgraph. So we have isometric subgraphs of order upto s − 2.
To remove s − 1 vertices, we delete every vertex of H0 except u. This is
again isometric since no geodesic from u to Gn − H0 goes through any
other vertex of H0. Next we remove u, which clearly leaves an isometric
subgraph. We can now continue to remove vertices in H1, H2, . . . , Hp in

29
that order using the same ideas as for removing the vertices of H0 and
always removing the vertices of Hi which were adjacent to Hi−1 ﬁrst.
This will produce an isometric subgraph of every order.
There remains to do the case r = 3. Since r is odd, n must be even so
let k = n/2. Let G have vertices u1, u2 . . . , uk and v1, v2, . . . , vk where the
two sets of vertices form paths in their given orders. Also add all edges
of the form ujvj except for u2, v2 and uk−1vk−1. Finally add the edges
u1v2, u2v1, uk−1vk, ukvk−1. The graph G12 will be found in the following
ﬁgure.
s
r
r
r
d
d
d
d

r
r d
d
d
d

r
pqr
r
r
pr
r
G12
=
u u u u u u
v v v v v v
1 2 3 4 5 6
1 2 3 4 5 6
We now just list the set of vertices to be removed. Also verifying that
the resulting subgraphs are isometric. We start with
{ u1}, {u1, uk}, {u1, v1, u2}, {u1, v1, u2, uk}, {u1, v1, u2, v2, uk}
and {u1, v1, u2, vk−1, uk, vk}.
We continue by deleting the 6 vertices in the last set and adding to them
the following vertices taken sequentially in the given order
u3, u4, . . . , uk−1, v2, v3, . . . , vk−2.
This completes the proof.
(c) GEODETIC GRAPH
A connected graph is called geodetic if there exists exactly one shortest
path between any two of its vertices. For example, every tree is a geode-
tic graph.

30
d
d
d

d
d
d

s r
s
r
r
rs s
r
v
v
v
v
v
v
1
4
5
7
8
6
v3
v2
T
uv
v
u
G
v
v v
1
2
v
3
4
BIGEODETIC GRAPH
Bigeodetic graphs are deﬁned as graphs in which each pair of vertices
has atmost two paths of minimum length between them. The class of
bigeodetic graphs contains all the geodetic graphs.
EXAMPLE
s
r
r
s
G
GENERALISED GEODETIC GRAPHS
It is a natural generalization of geodetic and bigeodetic graphs.
K-GEODETIC GRAPHS
Let G be a simple graph, that is without loops or multiple edges. We will
say G is a k-geodetic graph if each pair of vertices has at most k paths
of minimum length between them.
Remark
If a graph is k-geodetic then it will be p-geodetic with p ≥ k.
Proposition 3.0.4. If G is a k-geodetic graph then all its blocks are k-

31
geodetic.
Proof. If G is a k-geodetic graph then each pair of vertices has atmost
k shortest paths.
Let us suppose that there exists a block, B which is not k-geodetic. Then
there exists a block B which is not k-geodetic. Then there will be two
vertices x and y in B such that there will be k+l(l ≥ 1) paths of minimum
length between them. Now, in G there are k+l(l ≥ 1) paths of minimum
length between x and y. This contradicts the idea of G being a k-geodetic
graph. Hence the blocks of G must be k-geodetic.
Remark
The converse is not true.
The graph in the ﬁgure is not k-geodetic though the blocks of G are
k-geodetic, for k = 2
d
d
d
d
d
d
d
d
d
d

d
d
d
d
dd
s
s
rs
s
u
s
t
s
u
x yB1 B2
Theorem 3.0.5. A separable graph of diameter two is k-geodetic if, and
only if, G has exactly one cut vertex, all its blocks are k-geodetic of diam-
eter two at most and all the vertices of G are adjacent to the cut vertex
of G.
Proof. Let G be a separable k-geodetic graph of diameter two. Obviously
it should have exactly one cut vertex z, because otherwise the diameter
of G would be at least three. Since G is k-geodetic all its blocks are k-

32
geodetic. As the diameter of G is two then each block of G has diameter
two at most. Let x, y ∈ V (G) and let x ∈ V (B1) and y ∈ V (B2) where
B1 and B2 are two blocks of G. Since the diameter of G is two, both
vertices x and y must be adjacent to the cut vertex z.
Conversely, let G be a separable graph with exactly one cut vertex z,
all its k-geodetic blocks being of diameter two at most and all its vertices
adjacent to the cut vertex z. Then the diameter of G is two. If we choose
two nonadjacent vertices x, y ofG such that x, y ∈ V (Bi), 1 ≤ i ≤ k, then
as Bi is k-geodetic of diameter two at most, there will be at most k paths
of length two between x and y in Bi and hence in G. If x ∈ V (Bi), y ∈
V (Bj), i = j then d(x, y) = 2 since both are adjacent to the cut vertex
z, and this path is unique. Hence G is a separable k-geodetic graph of
diameter two.
The following ﬁgure shows a 4-geodetic graph of diameter two with three
blocks. This graph has a unique cut vertex and any vertex is adjacent to
the cut vertex.

s
u
s
r
s s
t
s
t
u
u
u
t t
s

33
Theorem 3.0.6. Let G be a separable graph where all its blocks are k-
geodetic and satisfy the following property: all the vertices of each block
Bi are adjacent to any cut vertex of Bi. Then G is k-geodetic.
Proof. If all its blocks are k-geodetic then the graph is at least k-geodetic.
We will prove that the geodeticity of the graph cannot be greater than
k. Let x, y be two vertices of the graph G.
case(i)
If x ∈ V (Bi) and y ∈ V (Bj), i = j then there is only one path between
x and y. That path is obtained as follows : the edge (x, zi), the path
linking zi with zj and the edge (zj, y) where zi and zj are cutvertices of
Bi and Bj respectively.
case(ii)
If x, y ∈ V (Bi) (the same block Bi )
In this case there are atmost k shortest paths between them because each
block is k-geodetic.
In both the cases there are atmost k shortest paths between the
vertices x and y.Since the vertices x and y are arbitrary, the graph is
k-geodetic.
Proposition 3.0.7. Given k, all the connected graphs of diameter two
with n ≤ k + 2 vertices are k-geodetic graphs. Besides, this number of
vertices is maximal.i.e., there exists at least a connected graph of diameter
two with n = k + 3 vertices which is not k-geodetic
Proof. We prove by induction. For k = 1, all the connected graphs of
diameter two with n ≤ 3 vertices are geodetic and the ﬁrst graph which is
not geodetic requires n = 4 vertices. This graph is the circuit graph C4.
For k = 2, all the connected graphs with n ≤ 4 vertices are bigeodetic and

34
it is possible to obtain a graph with n ≤ 5 vertices which is not bigeodetic.
G
x
1
s
rs
s r
Three paths of minimum length exists between x and y. For k = 3 all
the connected graphs with n ≤ 5 vertices are trigeodetic and the graph
on the following ﬁgure with n = 6 vertices is not trigeodetic .
d
d
d
d
r
rr
r
s
qpp
x
y
G2
Four paths of minimum length exist between x and y.
In general, we suppose all the connected graphs with n ≤ (k − 1) + 2
vertices are (k − 1)geodetic. If we add a new vertex and all necessary
edges to join that vertex with the remaining vertices, we will have a
new graph with n = k + 2 vertices. This graph has at most k paths of
minimum length between each pair of vertices. Therefore, this graph will
be k-geodetic. Besides for n = k + 3 it is always possible to build a new
graph which is not k-geodetic.
(d) INTERVAL DISTANCE MONOTONE GRAPH
A graph G is interval monotone if each interval in G is convex.

35
Definition 3.0.8. (Interval function)
Let G be a graph, and let P(V ) be the power set of V . The mapping
IG : V × V → P(V ) defined by
IG(u, v) := {w ∈ V |w lies on a shortest (u, v)-path in G} is the interval
function of G.
Definition 3.0.9. Let G be a graph with interval function I. A subset
W of V is convex if I(u, v) ⊆ W, for any u and V in W.
Definition 3.0.10. (Subdivision of an edge)
A subdivision of an edge e = uv of a graph G is obtained by introducing a
new vertex w in e, that is, by replacing the edge e = uv of G by the path
uwv of length 2 so that the new vertex w is of degree 2 in the resulting
graph.
Definition 3.0.11. (Homeomorph or a subdivision of a graph)
A homeomorph or a subdivision of a graph G is a graph obtained from G
by applying a finite number of subdivisions of edges in succession.G itself
is regarded as a subdivision of G.
Definition 3.0.12. (Homeomorphic graphs)
Two graphs G1 and G2 are called homeomorphic if they are both homeo-
morphs of some graph G
Definition 3.0.13. (Homomorphic Graphs)
A graph G is homomorphic to a graph H if there exists a homomorphism
from G onto H -that is, a mapping f from V (G) onto V (H) such that
if W ⊆ V (H) induces a connected subgraph of H, then the set f−1
[W]
induces a connected subgraph of G (or, informally, if we can obtain H
from G by contracting edges
EXAMPLES OF INTERVAL MONOTONE GRAPHS

36
r r r
rrq
v
v
v
v
v
1
2
3
4
5
T
Consider I(v2, v5) = {v2, v3, v4, v5}, I(v3, v4) = {v3, v4}
I(v3, v4) ⊆ I(v2, v5)
I(v1, v5) = {v1, v2, v3, v4, v5}, I(v2, v4) = {v2, v3, v4}
I(v2, v4) ⊆ I(v1, v5).
s qr
qp rp
v4
v v
v
1 2
3
G
I(v1, v2) = {v1, v2}, I(v1, v3) = {v1, v2, v3, v4}
I(v1, v4) = {v1, v4}
I(v1, v4) ⊆ I(v1, v3).
Proposition 3.0.14. If G contains no induced subgraph homomorphic
to K2,3 , or to the graph in the ﬁgure then G is interval monotone.

37
s
r
G
s
t
t
Fig 1
a
b
c
d
e
Proof. Let u and v be vertices of G for which the interval I(u, v) is
not convex. Choose x and y in I(u, v) with I(x, y) ⊆ I(u, v), and such
that d(x, y) is as small as possible. Note that d(x, y) ≥ 2 and that
x = u = y = v = x.
Since d(x, y) is as small as possible, ∃ a shortest (x, y)- path p = x →
p → . . . → y ∈ G, which has only the vertices x and y in common with
I(u, v).
Let q be a neighbor of x in I(x, u) and let Q = q → . . . → y be a
path from q to y in the subgraph of G induced by I(q, u) ∪ I(u, y) (the
following ﬁgure)
c
cd
c
d
d

dd
e
Q
P
p x
q
u
R
v
r y
Figure 2
(If we require Q to be a shortest (q, y)− path in this subgraph,then u does
not necessarily lie on Q ).Similarly,let r be a neighbour of x in I(x, v)
and let R = r → . . . → y be a path from r to y in the subgraph of G

38
induced by I(r, v) ∪ I(v, y).
Since p is not in I(u, v), the vertex p is adjacent to atmost one of the two
vertices q and r.
Let H be the subgraph of G induced by the vertices of P, Q and R.In
H we contract the edges in the subgraph H of H induced by the set
V (H) {x, p, q, r}, thus getting a graph H with vertices x, p, q, r and s,
where s is the homomorphic image of H . In H the vertex s is adjacent
to p, q and r but not to x. Hence it follows from the choice of x, p, q and
r that H is K2,3,or the graph in ﬁgure 1
Proposition 3.0.15. If G contains no subgraph homeomorphic to K2,3
,then G is interval monotone.
Proof. Let the vertices u, v, x and y and the paths P, Q and R be as in
the proof of the previous proposition.
If ∃ a vertex w in I(u, x) ∩ I(y, v), we would have x in I(w, v) ⊆ I(y, v)
and hence
I(x, y) ⊆ I(y, v) ⊆ I(u, v).
This contradicts the choice of x and y.So I(u, x) ∩ I(y, v) = ∅.
Similarly we have I(u, v) ∩ I(x, v) = ∅.
It follows that [I(u, x ∪ I(u, y)] ∪ [I(v, x) ∪ I(v, y)] = {x, y} so that the
paths Q and R have only the vertex y in common. This implies that P, Q
and R form a subgraph of G homeomorphic to K2,3,which completes the
proof.
(e) MEDIAN GRAPH
A connected graph G = (V, E) is called median, if for every three ver-
tices u, v, w ∈ V, ∃ a unique vertex that lies simultaneously on a shortest
(u − v)− path, a shortest (u − w)− path and a shortest (w − v)− path.

39
A graph G with interval function I is a median graph if |I(u, v, w)| = 1 for
any three vertices u, v and w of G, where I(u, v, w) := I(u, v) ∩ I(v, w) ∩
I(w, v).
u u
t
u
s t
u
u
u
su

4
4
4
4
4
d
d
d
d
v v
v
v
v
v
v
v
v
1
2
3
4
5
6
7
8
9
v
11
v
10
G
I(v1, v2) = {v1, v2}, I(v1, v3) = {v1, v2, v3}, I(v1, v4) = {v1, v2, v3, v4, v10, v11}
I(v1, v5) = {v1, v2, v3, v4, v5, v6, v10, v11}, I(v1, v6) = {v1, v2, v6, v10, v11}
I(v1, v7) = {v1, v6, v7, v8, v9, v10, v11}, I(v1, v2, v3) = {v3)}, I(v1, v2, v4) =
{v2)}
I(v2, v3) = {v2, v3}, I(v2, v4) = {v2, v3, v4, v11}, I(v2, v5) = {v2, v3, v4, v5, v6v11}
I(v2, v6) = {v2, v6, v11}, I(v2, v7) = {v2, v6, v7, v8, v11}, I(v2, v8) = {v2, v8, v11},
I(v2, v9) = {v1, v2, v8, v9, v10, v11}, I(v3, v4) = {v3, v4}, I(v3, v5) = {v3, v4, v5},
I(v3, v6) = {v3, v4, v5, v6, v11, v11},I(v3, v7) = {v3, v4, v5, v6, v7, v8, v11}, I(v3, v8) =
{v2, v3, v4, v8, v11}
I(v3, v9) = {v1, v2, v3, v8, v9, v10, v11}, I(v3, v10) = {v1, v2, v3, v10, v11}
I(v4, v5) = {v4, v5}, I(v4, v6) = {v4, v5, v6}, I(v4, v7) = {v4, v5, v6, v7, v8, v11}
I(v4, v8) = {v4, v8, v11}, I(v4, v9) = {v4, v8, v10, v9, v11}
I(v4, v10) = {v4, v10, v11}, I(v5, v6) = {v5, v6}, I(v5, v7) = {v5, v6, v7}
I(v5, v8) = {v4, v5, v6, v7, v8, v11}, I(v5, v9) = {v4, v5, v6, v7, v8, v9, v10, v11}

40
I(v5, v10) = {v4, v5, v6, v10}, I(v6, v7) = {v6, v7}, I(v6, v8) = {v6, v7, v8, v11}
I(v6, v9) = {v6, v7, v8, v9, v10, v11}, I(v6, v10) = {v6, v10, v11}, I(v7, v8) =
{v7, v8}
I(v7, v9) = {v7, v8, v9}, I(v7, v10) = {v6, v7, v8, v9, v10, v11}, I(v8, v9) = {v8, v9}, I(v8, v(
{v8, v9, v10, v11}
I(v9, v10) = {v9, v10}
I(v6, v7, v8) = I(v7, v6)∩I(v6, v8)∩I(v7, v8) = {v6, v7}∩{v6, v8}∩I(v7, v8)
= {v7}
Theorem 3.0.16. Let G be a median graph with interval function I.
Then I is a median interval structure on V
Proof. We have only to verify the interval monotonicity of I.
Assume the contrary, and let I(u, v) be a non-convex interval in G. Let
x and y be two vertices in I(u, v) such that I(x, y) ⊆ I(u, v) with d(x, y)
as small as possible. Not that d(x, y) ≥ 2. Then ∃ a shortest (x, y)-
path p such that all interval vertices of p lie outside I(u, v). Let z be
an interval vertex of p. It follows from the minimality of d(x, y) that
I(u, v) ∩ I(x, z) = x.
Since I(u, x, z) = ∅ it follows that x lies in I(u, z). Similarly x lies in
I(v, z) and so x ∈ I(u, v, z). i.e., x is the median of u, v and z.In the
same way it follows that y is also the median of u, v and z .So x = y ,
contradicting the fact that d(x, y) ≥ 2 .
Theorem 3.0.17. Let G be a graph with interval function I.If I is an
interval structure on V then G is a median graph.
Proof. Assume the contrary.Let u, v and w be vertices of G such that
I(u, v) ≥ 2.Then G is interval monotone ,we can choose two adjacent

41
vertices x and y in I(u, v, w) ≥ 2.Then since G is interval monotone,we
can choose two adjacent vertices x and y in | I(u, v, w) |≥ 2.
By the following proposition we get G is bipartite.
Proposition 3.0.18. Let G be a graph with interval function I such that
I(u, v, w) = ∅,for any three vertices u, v and w of G.Then G is connected
and bipartite.
Proof. Since any interval in G must be non-empty, G is connected.
Assume that G is not bipartite,and let u0 → u1 → . . . → u2k → uo be an
odd circuit of smallest length.
Then
d(u0, uk) = d(u0, uk+1) = k.
Hence uk+1 is not in I(u0, uk) and uk is not in I(u0, uk+1) so that
I(u0, uk, uk+1) = ∅, which is impossible- that is G is bipartite.
Hence d(u, x) and d(u, y) diﬀer by exactly 1, say
d(u, y) = d(u, x) + 1.
Since x and y lie in I(u, v) ∩ I(u, w), it follows that
d(y, v) = d(x, v) − 1
and
d(y, w) = d(x, w) − 1.
Hence since x and y lie in I(v, w) ,we have
d(v, w) = d(v, y) + d(y, w)
= d(v, x) + d(x, w) − 2
= d(v, w) − 2 , which is impossible.
Theorem 3.0.19. Let G be a connected graph with interval function I.
Then G is a median graph if and only if G is interval monotone and I

42
satisﬁes the following condition:
if I(u, v) ∩ I(v, w) = v then d(u, w) = d(u, v) + d(v, w) for u, v, w ∈ V
Proof. The ” only if ” part of the theorem follows from theorem 3.0.8
and the fact that if I(u, v) ∩ I(v, w) = v then v is the median of u, v and
w.
Conversely in view of theorem 3.0.9 it suﬃces to prove that I(u, v, w) = ∅
for any three vertices u, v and w of G .
Let u, v and w be arbitrary vertices of G and let z in I(u, v) ∩ I(v, w) be
such that
I(u, z) ∩ I(z, w) = z
Then d(u, w) = d(u, z) + d(z, w) -that is,z lies in I(u, w), and so z is in
I(u, v, w).

Conclusion
Distance metrics and distances have now become an essential tool in many ar-
eas of Mathematics and its applications including Geometry, Probability, Statistics,
Coding/Graph Theory, Clustering, Data Analysis, Pattern Recognition, Networks,
Engineering, Computer Graphics/Vision, Astronomy, Cosmology, Molecular Biol-
ogy and many other areas of science.
The concept of distances is widely used throughout graph theory and its appli-
cations. Distance is used in various graph operations, in isomorphism testing and
convexity problems and is the basis of several graph symmetry concepts. Distance
is used to deﬁne many graph centrality concepts, which in turn are useful in facility
location problems. Numerous graph algorithms are distance related. In that they
search for paths of various lengths within the graph. Distance is an important fac-
tor in extremal problems in graph connectivity. Graph connectivity is important
in its own right because of its strong relation to the reliability and vulnerability of
computer networks.
In the beginning of this project on distances in graph theory we have dis-
cussed shortest path distance, minimum weighted path distance, graph diameter,
steiner distance, truncated metric. Among them, shortest path distance, minimum
weighted path distance and truncated metric satisfy the conditions of metric and
the others do not. Then we went through path-quasi metric, strong metric, oriented
diameter and circular metric in digraphs. Among them strong metric and circular
43

44
metric satisfy the conditions of a metric. Path quasi metric is an example of a quasi
metric. Finally, we have studied some distance deﬁned graphs and their properties.
They are k-power of a graph, isometric subgraph of a graph, geodetic graph, interval
monotone graph and median graph.
In this project we have studied the notion of distances in graphs specifying nine
types of distances and ﬁve graphs related with one of these distances. We admit that
the study is not complete but surely it gave us an opportunity to get familiarised
with a new area in Graph Theory which is of great importance.

Bibliography
[1] Michel Maria Deza and Elena Deza (2009), Encyclopedia of Distances,
Springer
[2] R. Balakrishnan and K. Ranganathan (1999), A Text Book of Graph
Theory, Springer
[3] H. M. Mulder (1980), The Interval Function of a Graph, Mathematical
Centre Tracts
[4] R. M. Ramos, J. Sicilia, and M. T. Ramos (1998), A Generalization of
Geodetic Graphs: K-Geodetic graphs,Investigacion Operativa, Volume 6,
Numbers 1, 2 and 3, January-December 1998
[5] Dennis Ross, Bruce Sagan, Ronald Nussbaum, and Abdol-Hossein
Esfahanian (2014), On Constructing Regular Distance-Preserving
Graphs(Article), Cornell University Library
45

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