1. CS100: Discrete structures
Computer Science Department
Lecture 7: Graphs

2. Lecture Contents Types of Graphs Graph Terminology
Special Types of Graphs
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3. Simple Directed Graphs
Types of Graphs
Types of Graphs
undirected Graphs
Simple Graphs
Multigraphs
Pseudographs
directed Graphs
Simple Directed Graphs
Directed Multigraphs
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4. Undirected Graph A graph G = ( V , E ) consists of
V, a nonempty set of vertices (nodes)
E, a set of edges (unordered pairs of distinct elements of V)
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5. Types of undirected Graphs
Simple Graphs
Each edge connects two different vertices
No two edges connect the same pair of vertices
Multigraphs
Graphs that may have multiple edges connecting the same vertices
Pseudographs
Graphs that may include loops
Possibly multiple edges connecting the same pair of vertices
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6. Simple Graph Example
San Francisco
Denver
Los Angeles
New York
Chicago
Washington
Detroit
This simple graph represents a network that contains computers and telephone links between computers
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7. Multigraph Example
There can be multiple telephone lines between two computers in the network.
Detroit
New York
San Francisco
Chicago
Denver
Washington
Los Angeles
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8. Pseudograph Example
Detroit
Chicago
New York
San Francisco
Denver
Washington
Los Angeles
There can be telephone lines in the network from a computer to itself.
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9. Directed Graph A directed graph (digraph) ( V , E ) consists of
V, a nonempty set of vertices
E, a set of directed edges (arcs)
Each directed edge is associated with an ordered pair of vertices. The directed edge associated with the ordered pair (u,v) is said to start at u and end at v.
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10. Types of directed Graphs
Simple Directed Graph
No loops
No multiple directed edges
Directed Multigraph
Directed graphs that may have multiple directed edges from a vertex to a second (possibly the same)
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11. Simple Directed Graph Example
The edges are ordered pairs of (not necessarily distinct) vertices.
Detroit
Chicago
San Francisco
New York
Denver
Washington
Los Angeles
Some telephone lines in the network may operate in only one direction. Those that operate in two directions are represented by pairs of edges in opposite directions.
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12. Directed Multigraph Example
Detroit
New York
Chicago
San Francisco
Denver
Washington
Los Angeles
There may be several one-way lines in the same direction from one computer to another in the network.
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13. Graph Model Structure
Three key questions can help us understand the structure of a graph:
Are the edges of the graph undirected or directed (or both)?
If the graph is undirected, are the multiple edges present that connect the same pair of vertices?
If the graph is directed, are multiple directed edges present?
Are the loops present?
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14. Exercise Determine whether the graphs shown below have
directed or undirected edges,
multiple edges,
one or more loops A B
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15. directed or undirected
Solution
Graph
directed or undirected
multiple edges
Loops
Type A
undirected
multi-graph
no loops
Multigraph B
directed
directed multi-graph
two loops
Directed Multigraph
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16. Summary Type Edges Loops Multiple Edges Simple Graph Undirected NO
Multigraph
YES
Pseudograph
Simple Directed Graph
Directed
Directed Multigraph
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17. Precedence Graphs
In concurrent processing, some statements must be executed before other statements. A precedence graph represents these relationships.
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18. Graph Terminology
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19. Adjacent Vertices in Undirected Graphs
Two vertices, u and v in an undirected graph G are called adjacent (or neighbors) in G, if {u,v} is an edge of G.
An edge e connecting u and v is called incident with vertices u and v, or is said to connect u and v.
The vertices u and v are called endpoints of edge {u,v}.
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20. Degree of a Vertex
The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the degree of that vertex
The degree of a vertex v is denoted by deg(v)
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21. Example Find the degrees of all the vertices. Solution:b g e c d f
Find the degrees of all the vertices.
Solution:
deg(a)= , deg(b)= , deg(c)= , deg(d)= , deg(e)= , deg(f)= , deg(g)= .
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22. Adjacent Vertices in Directed Graphs
When (u,v) is an edge of a directed graph G,
u is said to be adjacent to (Neighbor) v .
v is said to be adjacent from u .
initial vertex terminal vertex
(u ,v)
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23. Degree of a Vertex In-degree of a vertex v Out-degree of a vertex v
The number of vertices adjacent to v (the number of edges with v as their terminal vertex)
Denoted by deg(v)
Out-degree of a vertex v
The number of vertices adjacent from v (the number of edges with v as their initial vertex)
Denoted by deg+(v)
A loop at a vertex contributes 1 to both the in- degree and out-degree.
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24. Degree of a Vertex : Example
Find the in-degrees and out-degrees of this digraph.
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25. Degree of a Vertex : Example
In-degrees: deg-(a) = 2, deg-(b) = 2, deg-(c) = 3, deg-(d) = 2, deg-(e) = 3, deg-(f) = 0
Out-degrees: deg+(a) = 4, deg+(b) = 1, deg+(c) = 2, deg+(d) = 2, deg+(e) = 3, deg+(f) = 0
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26. Theorem Let G = (V, E) be a graph with directed edges. Then:
The sum of the in-degrees of all vertices in a digraph = the sum of the out-degrees = the number of edges. (an edge is in for a vertex and out for the other)
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27. Exercise For following graph, determine:
The number of vertices and edges.
The in-degree and out-degree of each vertex for the given directed multi-graph.
The sum of the in-degrees of the vertices and the sum of the out- degrees of the vertices directly. Show that they are both equal to the number of edges in the graph.
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28. Solution
|V|= 4, |E|= 8.
The degrees are : deg−(a) =2, deg+(a) = 2, deg−(b) = 3, deg+(b) = 4, deg−(c) = 2, deg+(c) = 1, deg−(d) = 1,and deg+(d) = 1.
The sum of the in-degrees of the vertices =8, The sum of the out-degrees of the vertices =8, |E|= The sum of the in/out-degrees of the vertices = 8
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29. Complete graph
The complete graph on n vertices (Kn) is the simple graph that contains exactly one edge between each pair of distinct vertices.
The figures above represent the complete graphs, Kn , for n = 1, 2, 3, 4, 5, and 6.
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30. cycle
The cycle Cn (n  3), consists of n vertices v1, v2, …, vn and edges {v1,v2}, {v2,v3}, …, {vn-1,vn}, and {vn,v1}. C5 C6 C3 C4
Cycles:
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31. WHEEL
When a new vertex is added to a cycle Cn and this new vertex is connected to each of the n vertices in Cn, we obtain a wheel Wn. W5 W6 W3 W4
Wheels:
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32. Bipartite Graph
A simple graph is called bipartite if its vertex set V can be partitioned into two disjoint nonempty sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex in V2 (so that no edge in G connects either two vertices in V1 or two vertices in V2). b c a d e
a b
c d e
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33. Bipartite Graph : Example 1
1 2
5 4
Is C6 Bipartite?
Yes. Why?
Because:
its vertex set can be partitioned into the two sets V1 = {v1, v3, v5} and V2 = {v2, v4, v6}
every edge of C6 connects a vertex in V1 with a vertex in V2
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34. Bipartite Graph : Example 21
Is K3 Bipartite?
No. Why not?
Because:
Each vertex in K3 is connected to every other vertex by an edge
If we divide the vertex set of K3 into two disjoint sets, one set must contain two vertices
These two vertices are connected by an edge
But this can’t be the case if the graph is bipartite
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35. Exercise
Determine whether the graphs shown below are bipartite. Justify your answer. B A
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36. Solution
The graph is bipartite, its vertex set can be partitioned into two sets V1= { a, c } and V2 = { b, d, e }.
The graph is bipartite, its vertex set can be partitioned into two sets V1= { c, f } and V2 = { a, b, d, e }.
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37. Subgraph
A subgraph of a graph G = (V,E) is a graph H = (W,F) where W  V and F  E. C5 K5
Is C5 a subgraph of K5? …...
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38. Union
The union of 2 simple graphs G1 = (V1, E1) and G2 = (V2, E2) is the simple graph with vertex set V = V1V2 and edge set E = E1E2. The union is denoted by G1G2. S5 a b c e d f C5 b a c e d W5 b d a c e f
S5  C5 = W5
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39. Representing Graphs
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40. Adjacency Matrix
A simple graph G = (V , E ) with n vertices can be represented by its adjacency matrix A, where the entry aij in row i and column j is:
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41. Adjacency Matrix ExampleW5 d b a c e f
a b c d e f a b c d e f
From To
{v1,v2}
row column
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42. Incidence Matrix
Let G = (V,E) be an undirected graph. Suppose v1,v2,v3,…,vn are the vertices and e1,e2,e3,…,em are the edges of G.
The incidence matrix with respect to this ordering of V and E is the nm matrix M = [mij], where
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43. Incidence Matrix Example
Represent the graph shown with an incidence matrix.
e1 e2 e3 e4 e5 e6
edges
v v v3
v v5 e1 e2 e3 e4 e5 e6 v1 v2 v3 v4 v5
vertices
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44. Connectivity
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45. Paths in Undirected Graphs
There is a path from vertex v0 to vertex vn if there is a sequence of edges from v0 to vn
This path is labeled as v0,v1,v2,…,vn and has a length of m.
The path is a circuit if the path begins and ends with the same vertex.
A path is simple if it does not contain the same edge more than once.
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46. Paths in Undirected Graphs Cont.
A path or circuit is said to pass through the vertices v0, v1, v2, …, vn or traverse the edges e1, e2, …, en.
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47. Example 1 u1, u4, u2, u3 Is it simple? What is the length?
Does it have any circuits?
u u2
u5 u u3
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48. Example 2 u1, u5, u4, u1, u2, u3 Is it simple? What is the length?
Does it have any circuits?
u u2
u u3 u4
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49. Example 3 u1, u2, u5, u4, u3 Is it simple? What is the length?
Does it have any circuits?
u u2
u u3 u4
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50. Exercise
Given the following graph and list of vertices, for each list determine if :
It forms a path, if yes give the length
Is simple
any circuits
list of vertices
forms a path
length
simple
circuits
a,e,b,c,b.
e,b,a,d,b,e.
c,b,d,a,e,c.
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51. Connectedness Theorem
An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph.
Theorem
There is a simple path between every pair of distinct vertices of a connected undirected graph.
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52. Example Are the following graphs connected? …... …... c e f a d b g e
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53. Exercise ………… Determine whether the given graphs are connected.
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54. Connectedness (Cont.)
A graph that is not connected is the union of two or more disjoint connected subgraphs called the connected components of the graph.
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55. Example What are the connected components of the following graph? b d
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56. Example Cont.
What are the connected components of the following graph? b a c d e h g f
{a, b, c}, {d, e}, {f, g, h}
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57. Connectedness in Directed Graphs
A directed graph is strongly connected if there is a directed path between every pair of vertices.
A directed graph is weakly connected if there is a path between every pair of vertices in the underlying undirected graph.
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58. Example 1
Is the following graph strongly connected? Is it weakly connected?
This graph is strongly connected. Why? Because there is a directed path between every pair of vertices.
If a directed graph is strongly connected, then it must also be weakly connected.
a b c
e d
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59. Example 2
Is the following graph strongly connected? Is it weakly connected?
a b c
e d
This graph is not strongly connected. Why not?
Because there is no directed path between a and b, a and e, etc.
However, it is weakly connected. (Imagine this graph as an undirected graph.)
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60. Exercise weakly connected weakly connected
Determine whether each of these graphs is strongly connected and if not, whether it is weakly connected.
weakly connected
weakly connected
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61. Conclusion In this chapter we have covered:
Introduction to Graphs
Graph Terminology
Representing Graphs
Graph Connectivity
Refer to chapter 9 of the book for further reading.
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