10.2 graph terminology and special types of graphs · 10.2 graph terminology and special types of...

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10.2 Graph terminology and special types of graphs

[Def] Two vertices u and v in an undirected graph G are called adjacent (neighbors) in G if u and v are endpoints of an edge e of G.Such an edge is called incident with vertices u and v, and is said to connect u and v.

a

b c

d

adjacent vertices

incident with vertices a and c

connects u and v

2


[Def] neighborhood of v, N(v), is the set of all neighbors of v.

a

b c

d

adjacent vertices

3


[Def] neighborhood of v, N(v), is the set of all neighbors of v.

a

b c

d

adjacent vertices

neighborhood of a: N(a) = {b,c} ,neighborhood of b: N(b) = {a,d} ,neighborhood of c: N(c) = {a} ,neighborhood of d: N(d) = {b} ,

4


[Def] the degree of a vertex in 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.

denotaion: deg(v)

a

b c

d

5



denotaion: deg(v)

a

b c

d

deg(a) = 2 deg(c) = 1deg(b) = 2+2 = 4 deg(d) = 1

6



denotaion: deg(v)

a

b c

d

A vertex of degree 0 is called isolated.A vertex of degree 1 is called pendant.

pendant

pendant

7


[Theorem] The Handshaking TheoremLet G = (V,E) be an undirected graph with m edges.

Then 2m=∑v∈V

deg(v)

8



Then

a

b c

d

2m=∑v∈V

deg(v)

m = 4deg(a) = 2, deg(b) = 4, deg(c) = 1, deg(d) = 12 4 = 2 + 4 + 1 + 1 8 = 8

9



Then

Why is it so?

2m=∑v∈V

deg(v)

10



Then

Why is it so?

● each non-loop edge contributes 2 to the sum of the degrees ( 1 for each of adjacent vertices)

● each loop contributes 2 to the sum of the degrees (only for one vertex)

2m=∑v∈V

deg(v)

11



Then

[Theorem] An undirected graph has an even number of vertices of odd degree.

2m=∑v∈V

deg(v)

12



Then


Proof: Let V1 and V

2 be the set of vertices of even

degree and odd degree respectively, in a undirected graph G = (V,E) with m edges. |E| = m

2m=∑v∈V

deg(v)

13



Then


Proof: Let V1 and V



2m=∑v∈V

deg(v)

2m=∑v∈V

deg(v)=∑v∈V 1

deg(v)+∑v∈V 2

deg(v)

14



Then


Proof: Let V1 and V



2m=∑v∈V

deg(v)

2m=∑v∈V

deg(v)=∑v∈V 1

deg(v)+∑v∈V 2

deg(v)even

even

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Then


Proof: Let V1 and V



2m=∑v∈V

deg(v)

2m=∑v∈V

deg(v)=∑v∈V 1

deg(v)+∑v∈V 2

deg(v)even

must be even q.e.d.

even

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Let's discuss the terminology for directed graphs

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[Def] When (u,v) is an edge of the directed graph G, u is said to be adjacent to v, and v is adjacent from u.u is called initial vertex of (u,v), andv is called terminal/end vertex of (u,v).The initial and terminal vertex of the loop is the same.

a

b c

d

edges (a,b), (a,c):initial vertex,a is adjacent to b and to c

for edge (a,b):terminal vertex,b is adjacent from a;

for edge (b,d):initial vertex,b is adjacent to d

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[Def] The in-degree of a vertex v is the number of edges with v as their terminal vertex. deg-(v) The out-degree of a vertex v is the number of edges with v as their initial vertex. deg+(v)

Loop contributes to both in-degree and out-degree.

a

b c

d

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[Def] The in-degree of a vertex v is the number of edges with v as their terminal vertex. deg-(v) The out-degree of a vertex v is the number of edges with v as their initial vertex. deg+(v)

Loop contributes to both in-degree and out-degree.

a

b c

d

deg-(a) = 1 , deg+(a) = 3deg-(b) = 1 , deg+(b) = 1deg-(c) = 1 , deg+(c) = 0deg-(d) = 1 , deg+(d) = 0

20


[Theorem] Let G = (V,E) be a directed graph. Then

a

b c

d

deg-(a) = 1 , deg+(a) = 3deg-(b) = 1 , deg+(b) = 1deg-(c) = 1 , deg+(c) = 0deg-(d) = 1 , deg+(d) = 0 4 4|E| = 4

∑v∈V

deg−(v)=∑v∈V

deg+(v)=|E|

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a

b c

d

deg-(a) = 1 , deg+(a) = 3deg-(b) = 1 , deg+(b) = 1deg-(c) = 1 , deg+(c) = 0deg-(d) = 1 , deg+(d) = 0 4 4|E| = 4

∑v∈V

deg−(v)=∑v∈V

deg+(v)=|E|

Why is it so?

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a

b c

d

Because each edge has an initial vertex and a terminal vertex, hence contributes to both out-degree and in-degree.

∑v∈V

deg−(v)=∑v∈V

deg+(v)=|E|

Why is it so?

23


We can take a directed graph and convert to to undirected graph by ignoring the directions.Such a graph is called underlying undirected graph.

The directed graph and its underlying undirected graph have the same number of edges.

a

b c

d a

b c

d

24


Example: recall Hollywood graph.a) What does the degree of a vertex represent in the Hollywood graph?

b) What does the neighborhood of a vertex represent?

c) What do isolated and pendant vertices represent?

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the degree of a vertex represents the number of times the actor worked together with other actors on a movie or a TV show



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N(a) represents the list of all actors actor a worked with on a movie or a TV show.


27





Isolated vertices represent actors who didn't work with any other actor (present in the graph) on a movie or a TV show.Pendant vertices represent only actors with one collaboration only.

28


Example: show that in a simple graph with at least two vertices there must be two vertices that have the same degree.

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Solution: recall that simple graphs are undirected graphs that have no loops, no multiple edges.

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deg(v1) = 0, deg(v

2) = 0

v1 v

2

2 vertices:

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deg(v1) = 1, deg(v

2) = 1

v1 v

2

2 vertices:

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deg(v1) = 0, deg(v

2) = 0, deg(v

3) = 0,

v1 v

2

3 vertices:

v3

33




deg(v1) = 1, deg(v

2) = 1, deg(v

3) = 0,

v1 v

2

3 vertices:

v3

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deg(v1) = 2, deg(v

2) = 1, deg(v

3) = 1,

v1 v

2

3 vertices:

v3

35




deg(v1) = 2, deg(v

2) = 2, deg(v

3) = 2,

v1 v

2

3 vertices:

v3

36



v1 v

2

v1 v

2

v1 v

2

v3

v1 v

2

v3

v1 v

2

v3

v1 v

2

v3

...

...

...

37


Special Simple Graphs

- complete graphs ( Kn, for n Z+ )

- cycles ( Cn, for n Z+ and n 3)

- wheels ( Wn, for n Z+ and n 3)

- n-cubes ( Qn, for n Z+ )

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Special Simple Graphs: Complete graphs

A complete graph Kn on n vertices is a simple graph

that contains exactly one edge between each pair of distinct vertices.n = 1, 2, 3, 4, ...

K1 K

2 K3 K

4K

5

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Special Simple Graphs: Cycles

A cycle Cn on n vertices consists on n vertices v

1, v

2, …,

vn and edges {v

1,v

2}, {v

2,v

3},..., and {v

n-1,v

n}.

n = 3, 4, 5, ...

C3 C

4C

5 C6

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Special Simple Graphs: Wheels

A wheel Wn can be obtained from cycle C

n by adding

an additional vertex and connecting this new vertex with each of the n vertices in C

n.

n = 3, 4, 5, ...

W3 W

4W

5W

6

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Special Simple Graphs: n-Cubes

Graph of Qn has 2n vertices

Q1 Q

2 Q3

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n-Cubes can represent bit strings of length nn = 1, 2, 3, 4, ...

Q1 Q

2 Q3

43





Q1 Q

2

10Q

1

Q3

44





Q1 Q

2

Q2

10Q

1 00 01

10 11

2nd place

1st place

Q3

45





Q1 Q

2

Q2

10Q

1 00 01

10 11

2nd place

1st place

111110

000 001

011100

3rd place

1st place

2nd place

Q3

Q3

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Q1 Q

2 Q3

Q2

10Q

1 00 01

10 11

2nd place

1st place

111110

000 001

011100

3rd place

1st place

2nd place

010101

Q3

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Bipartite graphs

[Def] a simple graph is called bipartite if its vertex set V can be partitioned into two disjoint sets V

1 and V

2 such

that every edge in the graph connects a vertex in V1 and

a vertex in V2 (i.e. no same-set vertices connections).

We call the pair (V1, V

2) a bipartition of V.

Example:

a

b

c d

e

f a

b

c d

e

fbipartite graph not a bipartite graph

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Bipartite graphs


1 and V

2 such





Example:

a

b

c d

e

f a

b

c d

e

f

V1= {b,c,f}

V2 = {a,d,e}

49


Bipartite graphs


1 and V

2 such





Example:

a

b

c d

e

f a

b

c d

e

fbipartite graph not a bipartite graph

V1= {b,c,f}

V2 = {a,d,e}

50


Bipartite graphs

[Theorem] a simple graph is bipartite if and only if (iff) it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent vertics have the same color.

a

b

c d

e

fnot a bipartite graph

a

b

c d

e

fbipartite graph

51


Bipartite graphs

[Theorem] a simple graph is bipartite if and only if (iff) it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent vertices have the same color.

Proof: (): Assume that G = (V,E) is a bipartite graph, so there exist V

1 V and V

2 V, such that V

1 V

2 = ,

V = V1 V

2, and every edge in E connects a vertex from

V1 to vertex from V

2.

52


Bipartite graphs


Proof: (): Assume that G = (V,E) is a bipartite graph, so there exist V

1 V and V

2 V, such that V

1 V

2 = ,

V = V1 V

2, and every edge in E connects a vertex from

V1 to vertex from V

2.

Let's color all vertices from V1 with one color and all

vertices from V2 with other color. No adjacent vertices

will have the same color.

53


Bipartite graphs


Proof: (): Assume that it is possible to assign colors to the vertices of the graph using only two colors so that no two adjacent vertices have the same color.

54


Bipartite graphs


Proof: (): Assume that it is possible to assign colors to the vertices of the graph using only two colors so that no two adjacent vertices have the same color.Let V

1 be the set of vertices of color 1, and

V2 be the set of vertices of color 1, then

55


Bipartite graphs


Proof: (): Assume that it is possible to assign colors to the vertices of the graph using only two colors so that no two adjacent vertices have the same color.Let V

1 be the set of vertices of color 1, and

V2 be the set of vertices of color 1, then

V1 V

2 = , V = V

1 V

2, and every edge in E connects a

vertex from V1 to vertex from V

2, therefore, G is bipartite.

q.e.d.

56


Bipartite graphs

[Theorem] a simple graph is bipartite if and only if (iff) it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent vertics have the same color.

Theorem can be used to determine whether the graph is bipartite, and find the partition of bipartite graphs.

57


Bipartite graphs

Start with any vertex (with edge). Assign one color to it, say red.

a

b

c d

f

e

58


Bipartite graphs

Assign other color, say blue, to all vertices that are adjacent to it.

a

b

c d

f

e

59


Bipartite graphs

Then assign the red color to all vertices that are adjacent to vertices of blue color.

a

b

c d

f

e

60


Bipartite graphs

Done!V

1 = {a,d,e}, V

2 = {b,c,f}

a

b

c d

f

e

61


Bipartite graphs

Let's try the other one:

a

b

c d

e

f

62


Bipartite graphs

Let's try the other one: start with marking a red

a

b

c d

e

f

63


Bipartite graphs

Let's try the other one: continue by with marking vertices adjacent to a blue

a

b

c d

e

f

Problem: d and f are adjacent!!!

64


Bipartite graphs

Let's try the other one: continue by with marking vertices adjacent to a blue

a

b

c d

e

f

Problem: d and f are adjacent!!!

not a bipartite graph

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Complete bipartite graphs

A complete graph Kn on n vertices is a simple graph that

contains exactly one edge between each pair of distinct vertices. n = 1, 2, 3, 4, …

A complete bipartite graph Km,n

is a graph that has its vertex set partitioned into two subsets of m and n vertices, respectively with an edge between two vertices iff one vertex is in the first subset, and the other is in the second.

K1 K

2 K3 K

4K

5

66


Complete bipartite graphs

A complete bipartite graph Km,n

is a graph that has its vertex set partitioned into two subsets of m and n vertices, respectively with an edge between two vertices iff one vertex is in the first subset, and the other is in the second.

K2,3

K3,3

67


Bipartite graphs

Bipartite graphs can be used to model many types of applications that involve matching the elements of one set to elements of another.

Example: Suppose there are m employees in a group and n different jobs than need to be done, where m n.Each employee is trained to do one or more of these jobs. We want to assign one employee to each job.

Use bipartite graphs to model the situation!

68


Bipartite graphs

m employees n different jobs, where m n

Vertices: employees and jobsEdges: connect employees with jobs he/she can do

Smith Chen Yamagata Rodriguez Collins

requirements design implementationtesting

69


Bipartite graphs





A matching M is a set of edges from simple graph G, such that for any {s,t} and {u,v} M, s,t,u, and v are different vertices.

70


Bipartite graphs






71


Bipartite graphs






72


Bipartite graphs


A vertex that is an endpoint of an edge in matching M is said to be matched in M; otherwise it is said to be unmatched.

A maximum matching is a matching with the largest number of edges.

A complete matching M from V1 to V

2 is the one where

|M| = |V1|.

73


Bipartite graphs

Goal: one employee to each jobV

1 = jobs

V2 = employees



This is a complete matching, because|V

1| = 4 and we see 4 blue edges.

74


Bipartite graphs

Goal: one employee to each job, and all employees should get a jobV

1 = employees V

2 = jobs



This is not a complete matching, because|V

1| = 5 and we see only 4 blue edges.

75


Bipartite graphs

Example: marriages on an IslandSuppose that there are m men and n women on an island. Each person has a list of members of the opposite gender acceptable as a spouse.

76


Bipartite graphs


We construct a bipartite graph G = (V,E), where V1 is the set of

men, and V2 is the set of women. Edges are between people

that find each other acceptable as a spouse.

77


Bipartite graphs





A maximum matching is the largest possible set of couples to merry.

78


Bipartite graphs





A maximum matching is the largest possible set of couples to merry.A complete matching of set V

1 is a set of all couples where

every man is can merry, but possible not all women.

79


Bipartite graphs

Necessary and sufficient conditions for complete matchings:

[Theorem] Hall's marriage theoremThe bipartite graph G = (V,E) with bipartition (V

1,V

2) has a

complete matching from V1 to V

2 if and only if (iff)

|N(A)| |A| for all subsets A of V1.

No proof.

80


Bipartite graphs





A maximum matching is the largest possible set of couples to merry.A complete matching of set V

1 is a set of all couples where

every man is can merry, but possible not all women.

10.2 graph terminology and special types of graphs · 10.2 graph terminology and special types of...

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