Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph Theory
Math 301
Dr. Nahid Sultana
December 16, 2012
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graphs and Multigraphs
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Complete, Regular, and Bipartite graphs
Complete graph
Regular graph
Subgraphs, Isomorphic, and Homeomorphic graphs
Subgraphs
Isomorphic graphs
Homeomorphic graphs
Paths and Connectivity
Path
Simple Path
Cycle
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
A Graphs is a bunch of dots and lines where the lines connect
some pairs of dots.
Example:
I
The dots are called vertices and the lines are called edges.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
A Graphs is a bunch of dots and lines where the lines connect
some pairs of dots.
Example:
I
The dots are called vertices and the lines are called edges.
Types of graphs:
I
1. Simple graph (graph)
2. Multigraph
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Definition: A simple graph G (V , E ) consists of two things:
1. a finite set V of vertices
2. a finite set E of edges.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Definition: A simple graph G (V , E ) consists of two things:
1. a finite set V of vertices
2. a finite set E of edges.
I
Two vertices u and v are said to be adjacent if there is an
edge e = {u, v }.
Then u and v are called the endpoints of e.
The edge e is called the incident on each of its endpoints u
and v .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Definition: A simple graph G (V , E ) consists of two things:
1. a finite set V of vertices
2. a finite set E of edges.
I
I
Two vertices u and v are said to be adjacent if there is an
edge e = {u, v }.
Then u and v are called the endpoints of e.
The edge e is called the incident on each of its endpoints u
and v .
Example: Let V = {A, B, C , D} and E = {e1 , e2 , e3 , e4 }.
Where e1 = {A, D}, e2 = {A, C }, e3 = {B, D}, e4 = {B, C }.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Definition: A simple graph G (V , E ) consists of two things:
1. a finite set V of vertices
2. a finite set E of edges.
I
I
I
Two vertices u and v are said to be adjacent if there is an
edge e = {u, v }.
Then u and v are called the endpoints of e.
The edge e is called the incident on each of its endpoints u
and v .
Example: Let V = {A, B, C , D} and E = {e1 , e2 , e3 , e4 }.
Where e1 = {A, D}, e2 = {A, C }, e3 = {B, D}, e4 = {B, C }.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Now suppose we have another edge e5 = {B, C } as like above.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Now suppose we have another edge e5 = {B, C } as like above.
Then e4 and e5 are called Multiple edges, since they connect
same endpoints.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
I
I
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Now suppose we have another edge e5 = {B, C } as like above.
Then e4 and e5 are called Multiple edges, since they connect
same endpoints.
Suppose our graph has edge like e6 from a vertex D to itself.
This type of edge is called Loops:, since the endpoints are
same.
Multigraph: Diagram with multiple edges and/or loop.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
Definition: The degree of a vertex is the number of edges
having that vertex as an end point.
I
The degree of the vertex v is denoted by deg (v ).
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
Definition: The degree of a vertex is the number of edges
having that vertex as an end point.
I
The degree of the vertex v is denoted by deg (v ).
I
deg (1) =?, deg (5) =?, deg (8) =? in the following graph:
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
Definition: The degree of a vertex is the number of edges
having that vertex as an end point.
I
The degree of the vertex v is denoted by deg (v ).
I
deg (1) =?, deg (5) =?, deg (8) =? in the following graph:
I
A vertex of degree zero is called isolated vertex.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Theorem: The sum of the degrees of the vertices of a graph
G is equal to twice the number of edges in G .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
Theorem: The sum of the degrees of the vertices of a graph
G is equal to twice the number of edges in G .
I
Example:
Here deg (A) = 2, deg (B) = 3, deg (C ) = 3, deg (D) = 2.
Number of edges = 5.
Sum of the degrees equal 10, as expected.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
The previous theorem also holds for multigraphs, where a loop
is counted twice toward the degree of its endpoint.
I
Example:
Here deg (D) = 4, since the edge e6 is counted twice.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
Definition: A multigraph is said to be finite if it has a
finite number of vertices and a finite number of edges.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Graph
Simple graph
Multigraph
Degree of a vertex, isolated vertex
Finite graphs, Trivial graphs
I
Definition: A multigraph is said to be finite if it has a
finite number of vertices and a finite number of edges.
I
Definition: The finite graph with one vertex and no edges,
i.e., a single point, is called the trivial graph
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Complete graph
Regular graph
I
A graph G is said to be complete if every vertex in G is
connected to every other vertex in G .
I
The complete graph with n vertices is denoted by Kn
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Complete graph
Regular graph
I
A graph G is said to be complete if every vertex in G is
connected to every other vertex in G .
I
The complete graph with n vertices is denoted by Kn
I
Example:
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Complete graph
Regular graph
A graph G is regular of degree k or k − regular if every vertex
has degree k.
i.e. a graph is regular if every vertex has the same degree.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Complete graph
Regular graph
I
A graph G is regular of degree k or k − regular if every vertex
has degree k.
i.e. a graph is regular if every vertex has the same degree.
I
Example:
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Subgraphs
Isomorphic graphs
Homeomorphic graphs
A graph G1 (V1 , E1 ) is said to be a subgraph of a graph
G2 (V 2, E 2) if V1 ⊆ V2 and E1 ⊆ E2 .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Subgraphs
Isomorphic graphs
Homeomorphic graphs
I
A graph G1 (V1 , E1 ) is said to be a subgraph of a graph
G2 (V 2, E 2) if V1 ⊆ V2 and E1 ⊆ E2 .
I
Example:
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Subgraphs
Isomorphic graphs
Homeomorphic graphs
I
A graph G1 (V1 , E1 ) is said to be a subgraph of a graph
G2 (V 2, E 2) if V1 ⊆ V2 and E1 ⊆ E2 .
I
Example:
I
If v is a vertex in G , then G − v is the subgraph of G
obtained by deleting v from G and deleting all edges in G
which contain v .
I
If e is an edge in G , then G − e is the subgraph of G obtained
by simply deleting the edge e from G .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Subgraphs
Isomorphic graphs
Homeomorphic graphs
If G1 (V1 , E1 ) and G2 (V2 , E2 ) are two graphs, then we say that
G1 is isomorphic to G2 iff there exists a bijective function
f : V1 → V2 such that for every pair of vertices u, v ∈ V1 :
{u, v } ∈ E1 iff {f (u), f (v )} ∈ E2 :
The function f is called an isomorphism between G1 and G2 .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Subgraphs
Isomorphic graphs
Homeomorphic graphs
If G1 (V1 , E1 ) and G2 (V2 , E2 ) are two graphs, then we say that
G1 is isomorphic to G2 iff there exists a bijective function
f : V1 → V2 such that for every pair of vertices u, v ∈ V1 :
{u, v } ∈ E1 iff {f (u), f (v )} ∈ E2 :
I
I
The function f is called an isomorphism between G1 and G2 .
Example:
Here a corresponds to 1; b corresponds to 2; d corresponds to
4; c corresponds to 3.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
I
Subgraphs
Isomorphic graphs
Homeomorphic graphs
Two isomorphic graphs may be drawn very differently. For
example,
Proof in class.
Basic properties:
I
I
Isomorphic graphs must have the same number of vertices and
edges.
Same number of degrees at corresponding vertices.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Subgraphs
Isomorphic graphs
Homeomorphic graphs
Two graphs are said to be homeomorphic if and only if each
can be obtained from the same graph by adding vertices
(necessarily of degree 2) to edges.
Example:
Here the graphs (a) and (b) are homeomorphic since they can
be obtained from the graph (c) by adding appropriate vertices.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
A path in a graph/multigraph G to be any finite sequence of
vertices v0 , v1 , v2 , ..., vn and edges e1 , e2 , ..., en of G :
v0 , e1 , v1 , e2 , v2 , ..., vn−1 , en , vn ,
where the endpoints of ei are vi−1 and vi for each i.
I
We say it as the path from v0 to vn , or between v0 and vn , or
connects v0 to vn .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
A path in a graph/multigraph G to be any finite sequence of
vertices v0 , v1 , v2 , ..., vn and edges e1 , e2 , ..., en of G :
v0 , e1 , v1 , e2 , v2 , ..., vn−1 , en , vn ,
where the endpoints of ei are vi−1 and vi for each i.
I
We say it as the path from v0 to vn , or between v0 and vn , or
connects v0 to vn .
I
The length of a path is its number of edges.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
A path in a graph/multigraph G to be any finite sequence of
vertices v0 , v1 , v2 , ..., vn and edges e1 , e2 , ..., en of G :
v0 , e1 , v1 , e2 , v2 , ..., vn−1 , en , vn ,
where the endpoints of ei are vi−1 and vi for each i.
I
We say it as the path from v0 to vn , or between v0 and vn , or
connects v0 to vn .
I
The length of a path is its number of edges.
I
A closed path, sometimes called a circuit, is one in which
the first and last vertices are the same (i.e. v0 = vn ).
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
A simple path is a path in which all vertices are distinct.
I
A path in which all edges are distinct is called a trail.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
A simple path is a path in which all vertices are distinct.
I
A path in which all edges are distinct is called a trail.
I
Example:
P = 1 → 5 → 2 → 3 → 5 → 4.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
A simple path is a path in which all vertices are distinct.
I
A path in which all edges are distinct is called a trail.
I
Example:
P = 1 → 5 → 2 → 3 → 5 → 4.
I
Not a simple path as vertex 5 repeated but it is a trial
because none of the edges are repeated.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
A cycle is a closed path of length 3 or more in which all
vertices are distinct except v0 = vn .
I
Cn is a cycle of length n; Pn is a simple path of length n − 1.
I
Example:
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
Example:
P = 1 → 5 → 2 → 3 → 5 → 4.
Q=1→5→4.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
Example:
P = 1 → 5 → 2 → 3 → 5 → 4.
Q=1→5→4.
I
Here a path p exist from 1 to 4, and a simple path Q also
exist from 1 to 4.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
Example:
P = 1 → 5 → 2 → 3 → 5 → 4.
Q=1→5→4.
I
Here a path p exist from 1 to 4, and a simple path Q also
exist from 1 to 4.
I
Theorem: There is a path from a vertex u to a vertex v iff
there exists a simple path from u to v .
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
Two vertices u and v are connected to each other if there is
a path in the graph G which starts at u and ends at v .
I
The graph G is said to be connected if all vertices are
connected to each other.
I
Any vertex is connected to itself via the empty path.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
Two vertices u and v are connected to each other if there is
a path in the graph G which starts at u and ends at v .
I
The graph G is said to be connected if all vertices are
connected to each other.
I
Any vertex is connected to itself via the empty path.
I
The connected pieces of a graph are called its connected
components. Example:
Not connected but it has three connected component.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
The distance between vertices u and v in a graph G , written
d(u, v ), is the length of the shortest path between u and v .
I
The diameter of G , written diam(G ), is the maximum
distance between any two points in G .
I
Example:
Here d(A, F ) =? and diam(G ) =?
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
A vertex v in graph G is called a cutpoint if G − v is
disconnected.
I
An edge e of graph G is called a bridge if G − e is
disconnected.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Path
Simple Path
Cycle
Connectivity
Distance and Diameter
Cutpoints and Bridges
I
A vertex v in graph G is called a cutpoint if G − v is
disconnected.
I
An edge e of graph G is called a bridge if G − e is
disconnected.
I
Example:
In first figure,D is a cutpoint and there are no bridges.
In second figure,{D, F } is a bridge and D and F are cutpoints.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
Can you draw this graph without lifting your pen off the paper
and without retracing any edges?
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
Can you draw this graph without lifting your pen off the paper
and without retracing any edges?
I
yes.
3 → 6 → 5 → 2 → 6 → 4 → 5 → 1 → 2 → 3 → 4.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
Can you draw this graph without lifting your pen off the paper
and without retracing any edges?
I
yes.
3 → 6 → 5 → 2 → 6 → 4 → 5 → 1 → 2 → 3 → 4.
I
A graph like this is called drawable.
I
A graph G is called drawable or traversable if it is connected
and G contains a trial that uses every edges exactly once.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
Example: Is this graph drawable?
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
Example: Is this graph drawable?
I
No, This graph is not drawable.
If you want to draw every edges of this graph you have to
retrace an edge somehow.
I
How to prove this???
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
Key observation about drawable graph:
If G is drawable as a trial from u to v , then
G has to be connected,
and every vertex, except u and v , must have even degree.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
Key observation about drawable graph:
If G is drawable as a trial from u to v , then
G has to be connected,
and every vertex, except u and v , must have even degree.
I
Why?
Answer: If x 6= u and v .
Then every time we enter x with an edge we have to leave
this x with a different edge.
Because this is a trial and x is not the endpoint. So deg(x) =
even.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
Drawable
Not drawable
deg(1)=2, deg(2)=4, deg(3)=3,
deg(4)=3, deg(5)=4, deg(6)=4
deg(2)=3, deg(3)=3
deg(4)=3, deg(5)=3, deg(6)=4
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
Drawable
Not drawable
deg(1)=2, deg(2)=4, deg(3)=3,
deg(4)=3, deg(5)=4, deg(6)=4
deg(2)=3, deg(3)=3
deg(4)=3, deg(5)=3, deg(6)=4
I
Is the above drawable graph is closed trial?
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
The previous graph is not a closed trial because the starting
vertex and end vertex are not same.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
The previous graph is not a closed trial because the starting
vertex and end vertex are not same.
I
To draw a graph as a closed trial every vertex has to have
even degree.
I
Example:
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
The previous graph is not a closed trial because the starting
vertex and end vertex are not same.
I
To draw a graph as a closed trial every vertex has to have
even degree.
I
I
Example:
Theorem: A finite connected graph is Eulerian iff each
vertex has even degree.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
The previous graph is not a closed trial because the starting
vertex and end vertex are not same.
I
To draw a graph as a closed trial every vertex has to have
even degree.
I
Example:
I
Theorem: A finite connected graph is Eulerian iff each
vertex has even degree.
I
Every Eulerian graph is drawable but every drawable graph is
not Eulerian.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
The previous graph is not a closed trial because the starting
vertex and end vertex are not same.
I
To draw a graph as a closed trial every vertex has to have
even degree.
I
Example:
I
Theorem: A finite connected graph is Eulerian iff each
vertex has even degree.
I
Every Eulerian graph is drawable but every drawable graph is
not Eulerian.
I
Observation:If a graph is Eulerian, you can start and end at
any vertex, no special starting and ending points.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
A graph is Hamiltonian if it contains a cycle that goes
through every vertex.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
A graph is Hamiltonian if it contains a cycle that goes
through every vertex.
I
Eulerian graph: trail visits every edge exactly once, but may
repeat vertices.
I
Hamiltonian graph: cycle visits each vertex exactly once but
may repeat edges or may not use every edges.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
A graph is Hamiltonian if it contains a cycle that goes
through every vertex.
I
Eulerian graph: trail visits every edge exactly once, but may
repeat vertices.
I
Hamiltonian graph: cycle visits each vertex exactly once but
may repeat edges or may not use every edges.
I
There is no simple way to tell whether or not a graph is
Hamiltonian as like Eulerian graphs.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
I
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
A weighted graph is the same as a simple graph except that
we associate a real number (that is, the weight/ length) with
each edge in the graph.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
A weighted graph is the same as a simple graph except that
we associate a real number (that is, the weight/ length) with
each edge in the graph.
I
Example:
This a weighted graph where the weight of edge {a, b} is 5.
Math 301
Graph Theory
Outline
Graphs and Multigraphs
Complete, Regular, and Bipartite graphs
Subgraphs, Isomorphic, and Homeomorphic graphs
Paths and Connectivity
Euler and Hamilton Paths
Eulerian graph
Hamiltonian Graphs
Labeled and Weighted graph
I
A weighted graph is the same as a simple graph except that
we associate a real number (that is, the weight/ length) with
each edge in the graph.
I
Example:
This a weighted graph where the weight of edge {a, b} is 5.
I
The weight (or length) of a path in a weighted graph defined
to be the sum of the weights of the edges in the path.
Math 301
Graph Theory