Introduction to Graph Theory

Graph -Theory
by
Manash Kumar Mondal
M.Tech(CSE)
University of Kalyani

Outline
• Definition
• Finite and infinite graphs
• Directed and undirected graphs
• Degree
• Isolated vertex
• Pendent vertex
• Walks
• Null graphs
• Path
• Circuit
• Connected and disconnected
graph
• Euler's graph
• Hamiltonian path and circuit
• Trees
8/6/2018 Manash Kumar Mondal 2

• A graph G is a tuple consisting of a finite set V of vertices and a finite set E of
edges where each edge is an unordered pair of vertices. The two vertices associated
with an edge e are called the end-vertices of e. We often denote by (u,v), an edge
between two vertices u and v. We also denote the set of vertices of a graph G by
V(G) and the set of edges of G by E(G).A vertex of a graph is also called as a node
of a graph.
• Example-
Definition of graph

Multigraph
• A loop is an edge whose end-vertices are the same.
• Multiple edges (or parallel edges) are edges with the same pair of end-vertices.
• If a graph G does not have any loop or multiple edge, then G is called a simple
graph; otherwise, it is called a multigraph.
• A simple graph has no loop or multiple edge.

• A graph with a finite number of nodes and edges. If it has n nodes and has no
multiple edges or graph loops (i.e., it is simple), it is a subgraph of the complete
graph Kn.
• A graph which is not finite is called infinite. If every node has finite degree, the
graph is called locally finite. The Cayley graph of a group with respect to a finite
generating set is always locally finite, even if the group itself is infinite.
Finite and infinite graphs

• A graph is called directed graph or digraph if each edge is associated
with a direction.
• An undirected graph is a graph where each edge is directed in both
directions.
Undirected
graphs
Directed graphs
Directed and Undirected Graphs

• Adjacency and incident :Let e = (u,v) be an edge of a graph G. Then
the two vertices u and v are said to be adjacent in G, and the edge e is
said to be incident to the vertices u and v. The vertex u is also called a
neighbor of v in G and vice versa.
• Example-In the graph in
the vertices v1 and v3 are
adjacent; the edge e is incident
to the vertices v1 and v3 .The
neighbors of the vertex v1 in G are v2 , v3, v6, v9,and v11.
Adjacency and incident

Degree of a vertex
• The degree of a vertex v in a graph G, denoted by deg(v) or d(v), is the
number of edges incident to v in G, with each loop at v counted twice.
• Since the degree of a vertex counts its incident edges, it is obvious that
the summation of the degrees of
all the vertices in a graph is related
to the total number of edges
in the graph.
Degree is 3

(Degree-sum Formula)
• Let G = (V, E) be a graph with m edges. Then 𝑣∈𝑉 deg(𝑣) =
2m.
• Solution: Every non loop edge is incident to exactly two distinct vertices
of G. On the other hand, every loop edge is counted twice in the degree
of its incident vertex in G. Thus, every edge, whether it is loop or not,
contributes a two to the summation of the degrees of the vertices of G.

Isolated vertex and Pendent vertex
• An isolated vertex is a vertex with degree zero; that is, a vertex that is
not an endpoint of any edge (the example image illustrates one isolated
vertex).
• A vertex of a graph is said to be pendant if its neighborhood contains
exactly one vertex. A leaf vertex (also pendant vertex) is a vertex with
degree one.
Pendent vertex
Isolated vertex

Null Graphs
• An empty graph on n nodes consists of n isolated nodes with no edges.
Such graphs are sometimes also called edgeless graphs or null graphs
(though the term "null graph" is also used to refer in particular to the
empty graph on 0 nodes).
• The empty graph on 0 nodes is called the null graph, and the empty
graph on 1 node is called the singleton graph. The empty graph on n
vertices is the graph complement of the complete graph K_n, and is
commonly denoted K^n.

Regular graphs
• If all the vertices of a graph G have equal degrees, then we call G a
regular graph. We call it a k-regular graph if the common degree is k.
• Only the graphs with empty edge sets(having non empty vertex sets) are
0-regular.These are called null graphs.
• A 1-regular graph consists of a set of edges such that no two edges are
incident to a common vertex.
• A graph which consists of a list of vertices such that each pair of
consecutive vertices are adjacent with each other and the first vertex is
also adjacent with the last vertex, is a 2-regular graph. Such a graph is
called a cycle or a simple cycle. A graph which is a collection of simple
cycles is also a 2-regular graph.

Regular Graphs(cont.)
• A 3-regular graph is also called a cubic graph. Each vertices has degree 3.
2 -regular
3 –regular
Peterson graph
4-regular 5 -regular

Regular Graphs(cont.)
• Corollary : Every regular graph with an odd degree has an even number
of vertices.
• Corollary : A k-regular graph with n vertices has nk/2 edges.

SubGraph
• A subgraph of a graph G = (V, E) is a graph G ‘ = (V ‘ , E ‘ ) such that
V ‘ ⊆ V and E ‘ ⊆ E.
• We can obtain subgraphs of a graph G by deleting some vertices and
edges of G.
Graph G Subgraph of G Subgraph of G
Subgraph of G Subgraph of G

Complete Graphs
• A graph in which each pair of distinct vertices are adjacent is called a
complete graph. A complete graph with n vertices is denoted by Kn . It
is trivial to see that Kn contains n(n − 1)/2 edges.
• Example- A complete graph K6 with six vertices.
Complete Graph (K8)
Complete Graph (K6)

Independent Set and Bipartite Graphs
• Let G = (V, E) be a graph. A subset of vertices V’ ⊆ V is called an
independent set in G if for every pair of vertices u, v ∈ V’ , there is no
edge in G joining the two vertices u and v.
• A graph G is called a bipartite graph if the vertex set V of G can be
partitioned into two disjoint nonempty sets V 1 and V 2 , both of which
are independent. The two sets V 1 and V 2 are often called the partite
sets of G.
• 2 𝑛
− 2 possible bipartition of a graph with n vertices.

Bipartite Graphs
• Example-
Independent set Independent setIndependent set
Independent set

Path Graphs
• A path graph is a graph G that contains a list of vertices v 1 , v 2 , ..., v p
of G such that for 1 ≤ i ≤ p−1, there is an edge (v i ,v i+1 ) in G and
these are the only edges in G. The two vertices v 1 and v p are called the
end-vertices of G. A path graph with n vertices is denoted by Pn .
• Note that the degree of each vertex of a path graph is two except for the
two end-vertices, both of which have degree one
• Example- a path graph with six vertices
Path Graph

Cycle Graphs and Wheel Graphs
• A cycle graph is one that is obtained by joining the two end-vertices of a
path graph. Thus, the degree of each vertex of a cycle graph is two. A
cycle graph with n vertices is often denoted by Cn .
• A wheel graph with n vertices, denoted by W n , is obtained from a cycle
graph Cn−1 with n − 1 vertices by adding a new vertex w and joining an
edge from w to each vertex of Cn−1
Cycle graph
Wheel graph

Walks
• Let G be a graph. A walk in G is a nonempty list W = v 0 , e1 , v1 , ..., v
f−1 , ef ,vf , whose elements are alternately vertices and edges of G
where for 1 ≤ i ≤ f ,the edge e i has end vertices vi−1 and v i . The
vertices v 0 and v f are called the end vertices of W. If the end vertices
of a walk W of a graph G are u and v respectively, W is also called an u,
v-walk in G.
• Example-W = a,(a,i),i,(i,h),h,(h,c),c,(c,b),b
is a walk. W is also an a, b-walk in G
since the end vertices of W are a and b.

Trails
• A trail of a graph G is a walk in G with no repeated edges. That is, in a
trail an edge can not appear more than once.
• The walk W1 = a,(a,i),i,(i,h),h,(h,c),c,(c,b),b is a trail but the walk W2 =
a,(a,i),i,(i,h),h,(h,c),c,(c,b),b,(b,a), a,(a,i) is not a trail since the edge (a,i) has
appeared in W2 more than once. A trail is called a circuit when the two
end vertices of the trail are the same.
• A trail is called a circuit when the
two end vertices of the trail are the same.

Paths
• A path is a walk with no repeated vertex (except end vertices). When the
end vertices repeat, then it is called a closed path.
• In this a,(a,i),i,(i,h),h,(h,c),c,(c,b),b,(b,a),a is a closed path.
• A closed path is called a cycle. A
u,v-path is a path whose end
vertices are u and v.

Cycles
• A closed path is called a cycle. A u,v-path is a path whose end vertices
are u and v.
• A vertex on a path P which is not an end vertex of P is called an internal
vertex of P. The length of a walk, a trail, a path, or a cycle is its number
of edges. Thus a path of n vertices has length n − 1, and a cycle of n
vertices has length n.

Connected and disconnected graph
• A graph is connected when there is a path between every pair of vertices.
In a connected graph, there are no unreachable vertices.
• A graph that is not connected is disconnected. A graph G is said to
be disconnected if there exist two nodes in G such that no path in G has
those nodes as end points. A graph with just one vertex is connected.
An edgeless graph with two or more vertices is disconnected.
• Example-With vertex 0, this graph is
disconnected. The rest of the graph
is connected.

Connected and disconnected
graph(cont.)
Components: A maximal connected subgraph of G is a subgraph that is
connected and is not contained in any other connected subgraph of G. A
maximal connected subgraph of G is called a connected component of G.
• A directed graph is called weakly connected if replacing all of its directed
edges with undirected edges produces a connected (undirected) graph. It
is connected if it contains a directed path from u to v or a directed path
from v to u for every pair of vertices u, v. It is strongly connected, di-
connected, or simply strong if it contains a directed path from u to v and
a directed path from v to u for every pair of vertices u, v. The strong
components are the maximal strongly connected subgraphs.

Connected and disconnected
graph(cont.)

Eulerian Graphs
• In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite
graph which visits every edge exactly once.
• An Eulerian circuit or Eulerian cycle is an Eulerian trail which starts and
ends on the same vertex.
• Example-
(a) An Eulerian graph
(b) a graph which is not
Eulerian

Theorem(Euler graph)
• A connected graph G is Eulerian if and only if every vertex of G
has even degree.
Proof: Necessity: Suppose G is Eulerian. We wills how that every vertex
ofG has even degree. Since G is Eulerian, G contains an Eulerian circuit,
say, v0 ,e1 ,v1 ,e2 ,...,en ,v0 . Both edges e1 and en contribute 1 to the
degree of v 0 ; so degree of v 0 is at least 2.Each time the circuit passes
through a vertex (including v 0 ), the degree of the vertex is increased by
two. Therefore, the degree of every vertex including v 0 is an even integer.
Sufficiency: Assume that every vertex of G has even degree. We will
show that G is Eulerian, that is, G contains an Eulerian circuit.

Theorem(Euler graph)(cont.)
We give a constructive proof.
Let v0 be an arbitrary vertex. Starting from v0 , we find a maximal trail T. Since
every vertex of G has even degree, T will be a closed trail. If T contains all the edges
of G, then G is Eulerian. Otherwise, let G’ be the graph obtained from G by deleting
all the edges of T (see Fig next page). Since T has even degree at every vertex, every
vertex of G’ has even degree. Since G is connected, some edge e of G’ is incident to
a vertex v of T. We find a maximal trail T’ in G’ starting from v along e. Combining
T and T’ we get a longer circuit. Let T = T ∪ T’ and we repeat the argument.
Since E(G) is finite, this process must end, and it can only end by constructing
an Eulerian circuit.

Theorem(Euler graph)(cont.)

Hamiltonian Graphs
• Let G be a graph. A path in G that includes every vertex of G is called a
Hamiltonian path of G. A cycle in G that includes every vertex in G is
called a Hamiltonian cycle of G. If G contains a Hamiltonian cycle,then
G is called a Hamiltonian Graph.
• Every Hamiltonian cycle of a Hamiltonian graph of n vertices has
exactly n vertices and n edges. If the graph is not a cycle, some edges of
G are not included in a Hamiltonian Cycle.

Hamiltonian Graphs(cont.)
Non -Hamiltonian
Hamiltonian

Connectivity
• The connectivity κ(G) of a connected graph G is the minimum number of vertices
whose removal results in a disconnected graph or a single vertex graph K 1 . A graph
G is k-connected if κ(G) ≥ k. A separating set or a vertex cut of a connected graph
G is a set S ⊂ V(G) such that G − S has more than one component. If a vertex cut
contains exactly one vertex, then we call the vertex cut a cut vertex. If a vertex cut
in a 2-connected graph contains exactly two vertices, then we call the two vertices a
separation-pair.

Tree
• A tree is a connected graph that contains no cycle.
• A path is always a tree.
• A path with n vertices has n − 1 edges.
• In fact, every tree with n vertices has n − 1 edges
• A vertex with degree one in a tree T is called a leaf of T. All the vertices
of T other than the leaves are called the internal vertices of T.
• A collection of trees is called a forest. In other words, a forest is a graph
with no cycle. Such a graph is also called an acyclic graph. Each
component of a forest is a tree.

Tree(cont.)

Properties of a Tree
• Every tree with two or more vertices has at least two leaves.
• Every edge in a tree is a cut edge.
• Let G be a graph with n vertices. Then, any two of the following
three statements imply the third (and characterize a tree of n
vertices).
• (a) G is connected.
• (b) G contains no cycle.
• (c) G has n − 1 edges.
• A simple graph G is a tree if and only if for each pair of vertices u
and v of G, there is a unique u,v-path in G

Rooted Tree
• A rooted tree is a tree in which one of the vertices is distinguished from
the others.
• The distinguished vertex is called the root of the tree. The root of a tree
is usually drawn at the top.

Rooted Tree
In this fig. the root is v1 . If a rooted
tree is regarded as a directed graph in
which each edge is directed from top to
bottom, then every vertex u other than
the root is connected by an edge from
some other vertex p, called the parent of
u. We also call u a child of vertex p.

Spanning Trees of a Graph
• A subgraph G‘ of a graph G is a spanning subgraph of G if G’ contains
all vertices of G. A spanning subgraph T of a graph G is a spanning tree
of G if T is a tree.

Spanning Trees of a Graph(cont.)
• Let G = (V, E)be a connected graph and let S ⊆ E such that G−S is
disconnected. Then every spanning tree of G has an edge in S.
Let G be a connected graph. If G has no cycle, then G itself a spanning tree
of G. We thus assume that G has cycles. We delete edges from cycles one by one
until the resulting graph G ‘ is acyclic. Since no edge on a cycle is a cut edge by
Lemma 3.1.4(An edge is a cut-edge if and only if it belongs to no cycle.), G’ is
connected. Hence, G’ is a desired spanning tree T of G.’
Based on the proof of Lemma 4.4.1, one can develop an algorithm for finding a
spanning tree of a connected graph as follows: Check whether the graph has a cycle
or not. If the graph has a cycle, delete an edge from the cycle and repeat the process
until the resulting graph is acyclic.

Introduction to Graph Theory

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Introduction to Graph Theory