§8.1 Introduction to Graphs

Discrete

Mathematics

Chapter-8

Graphs

§8.1 Introduction to Graphs

„ Def 1. A (simple) graph G=(V,E) consists of a nonempty set V of vertices, and E, a set of unordered pairs of distinct elements of V called edges.

„ eg.

V₁

V₂

V₃ V₄

V₇

V₆

G=(V,E), where V={ v₁,v₂,…,v₇ }

E={ {v₁,v₂}, {v₁,v₃}, {v₂,v₃} {v₃,v₄}, {v₄,v₅}, {v₄,v₆} {v₄,v₇}, {v₅,v₆}, {v₆,v₇} }

V₅

„ Def 2. Multigraph

simple graph + “兩點間允許多條邊”

multiedge

eg.

V1

V2

V3 V4

V5

V7

V6

„ Def 3. pseudo graph :

simple graph + multiedge + loop

( loop即 )

eg.

„ Table 1. Graph Terminology

Type Edges Multiple

Edges

Loops (simple) graph

Multigraph

Pseudo graph Directed graph

Directed multigraph

Directed

表示成 (u,v) Undirected 表示成 {u,v}

„ Note: directed multigraph 中

U V

是 multiedge 邊為(u,v),(u,v)

U V

不是 multiedge 邊為(u,v),(v,u)

Exercise : 3,5,6,7,9

§8.2 Graph Terminology

„ Def 1. Two vertices u and v in a graph

G are called adjacent in G if {u,v} is an edge of G.

„ Def 2. The degree of a vertex v, denoted

by deg(v), in an (undirected) graph is the number of edges incident with it.

(undirected)

(Note : 點跟點相連 : adjacent 點跟邊相連 : incident )

(Note : loop要算2次)

„ Example 1. What are the degree of the vertices in the graph H ?

„ Sol :

a b

c

e d f

H

deg(a)=4 deg(b)=6 deg(c)=1 deg(d)=5 deg(e)=6 deg(f)=0

„ Def. A vertex of degree 0 is called isolated.

eg. “ f ” in Example 1.

Thm 1. (The Handshaking Theorem)

Let G=(V,E) be an undirected graph with n edges (i.e., |E|=n).

Then

∑

∈

=

V v

n v ) 2 deg(

Pf : 每條edge {u,v}會貢獻一個degree給u跟v

„ eg. Example 1. there are 11 edges, and

∑

=

V v

v ) 22 deg(

Example 2. How many edges are there in a graph with 10 vertices each of degree 6 ?

Sol :

10 ⋅ 6 = 2n => n=30

„ Thm 2. An undirected graph has an even number of vertices of odd degree.

„ Def 3. G: directed graph , G=(V,E) (u,v)∈E : u is adjacent to v

v is adjacent from u

u : initial vertex , v : terminal vertex

u v

„ Def 4.

G=(V,E) : directed graph, v∈V

deg^-(v) : # of edges with v as a terminal.

(in-degree)

deg⁺(v) : # of edges with v as a initial vertex (out-degree)

„ Example 3.

a b

c

e d

f

a is adjacent to b , b is adjacent from a a : initial vertex of (a,b)

b : terminal vertex of (a,b)

deg^-(a)=2, deg⁺(a)=4 deg^-(b)=2, deg⁺(b)=1

: :

deg^-(f)=0, deg⁺(f)=0

„ Thm 3. Let G=(V,E) be a digraph. Then

pf :

每個 edge貢獻一個 out–degree 給 a

一個 in–degree 給 b

a b

∑ ∑

∈ ∈

+

−

= =

V

v v V

E v

v ) deg ( ) (

deg

Def : The complete graph on n vertices, denoted by K_n, is the simple graph that contains

exactly one edge between each pair of distinct vertices.

Example 4.

K₁ K₂ K₃ K₄

Note : | E | =

n 2

„ Example 5. The cycle C_n, n≧3, consists of n vertices v₁,v₂,…,v_n and edges

{v₁,v₂}, {v₂,v₃},…,{v_n-1,v_n},{v_n,v₁}.

C₅ C₆

| E | = n

„ Example 6. W_n : (wheel), C_n中加一點連至 其餘n點(n≧3)

W₅ W₆

| V | = n + 1

| E | = 2n

„ Def 5. A simple graph G=(V,E) is called

bipartite if V can be partitioned into V₁ and V₂, V₁∩V₂=∅, such that every edge in the graph connect a vertex in V₁ and a vertex in V₂.

Example 8.

v1 v3

v5

v2 v4 v6

∴ C₆ is bipartite.

Example 10. Is the graph G bipartite ?

a b

g

f

e

d c

a

b

g f e

d

c

Yes !

„ Example 11. Complete Bipartite graphs (k_m,n) ???

K_2,3 K_3,3

Note. | E | = mn

„ Example 14. A subgraph of K₅

Def 6. A subgraph of a graph G=(V,E) is a

graph H=(W,F) where W ⊆ V and F ⊆ E.

(注意 F 要連接 W 裡的點)

a

b

d c e

a

e b

c

subgraph of K₅ K₅

„ Def 7. The union of two simple graph G₁=(V₁,E₁) and G₂=(V₂,E₂) is the

simple graph G₁∪G₂=(V₁∪V₂,E₁∪E₂)

„ Example 15.

a b c

d f

a b c

d e

a b c

d e f

G₁ G₂

G₁∪G₂

ex35上面 A simple graph G=(V,E) is called regular if every vertex of this graph has the same degree. A regular graph is called

n-regular if deg(v)=n , ∀v∈V.

eg.

K₄ : is 3-regular.

Exercise : 5,7,21,23,25,35,37

§8.3 Representing Graphs and Graph Isomorphism

※Adjacency list

Example 1. Use adjacency list to describe the simple graph given below.

b a

e d

c

Vertex Adjacent vertices

a b,c,e

b a

c a,d,e

d c,e

e a,c,d

Sol :

„ Example 2. (digraph)

a

b

e

c

d

Initial vertex Terminal vertices

a b,c,d,e

b b,d

c a,c,e

d

e b,c,d

※Adjacency Matrices

Def. G=(V,E) : simple graph,

V={v₁,v₂,…,v_n}. (順序沒關系)

A matrix A is called the adjacency matrix of G if A=[a_ij]_nxn , where

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

0 0

1 1

0 0

1 1

1 1

0 1

1 1

1 0

A1

Example 3.

a

c

b

d

undirected graph 的連 通矩陣必

“對稱”

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

0 1

1 1

1 0

0 1

1 0

0 1

1 1

1 0

2 A

a b c d

a b c d b d c a

b d c a

Example 5. (Pseudograph) (矩陣未必是0,1矩陣.)

Def. If A=[a_ij] is the adjacency matrix for the directed graph, then

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

0 2 1 2

2 1 1 0

1 1 0 3

2 0 3 0 A

a b c d

a b c d

a

=

1 , if

0 , otherwise

故矩陣 未必對稱

a b

c d

v_i v_j

„ Def 1.

The simple graphs G₁=(V₁,E₁) and G₂=(V₂,E₂) are isomorphic if there is an one-to-one and onto function f from V₁ to V₂ with the property that a~b in G₁ iff f(a)~f(b) in G₂, ∀a,b∈V₁

f is called an isomorphism.

※Isomorphism of Graphs

u₁

u₃

u₂

u₄

G

v₁

v₃

v₂

v₄

H

G is isomorphic to H

„ Example 8.

u₁

u₃

u₂

u₄

G

v₁

v₃

v₂

v₄

H

f(u₁) = v₁ f(u₃) = v₃ f(u₂) = v₄ f(u₄) = v₂

※Isomorphism Graphs 必有 : (1) 相同的點數。

(2) 相同的邊數。

(3) 相同的degree分佈。

※給定二圖，判斷它們是否isomorphic的問題一般來說不 易解，而且答案常是否定的。

Example 9.

Show that G and H are not isomorphic.

a

e

b

c

d

b

a

e d

c

Sol :

∵ H 有 degree = 1 的點，G 沒有

∴ G ≈ H

G H

Example 10.

Determine whether G and H are isomorphic.

G H

a b

d c

e

h

f g

s

v

t

u

w x

z y

Sol :

∵ G 中 degree 為 3 的點有d, h, f, b 它們不能接成 4-cycle

但 H 中 degree 為3的點有s, w, z, v 它們可接成 4-cycle

∴ 不是 isomorphic.

另法 : G 中 degree 為 3 的點，旁邊都只連了另一個 deg = 3 的點 但 H 中 deg = 3 的點旁邊都連了 2 個 deg = 3 的點。

Example 11. Show that G H

≈

G H

u₁ u₂

u₄ u₃

u₅

u₆

v₁

v₂

v₃

v₄

v₅ v₆

Exercise : 3,7,14,17,19,23,37,39

(1) 相同的點數。

(2) 相同的邊數。

(3) 相同的degree分佈

§8.4: Connectivity

Def. 1,2 :

„ In an undirected graph, a path of length n from u to v is a sequence of adjacent vertices going from vertex u to vertex v. (e.g., P: u=x₀, x₁,

x₂, …, x_n=v)

„ Note. A path of length n has n+1 vertices，n edges

„ A path is a circuit if u=v.

„ A path traverses the vertices along it.

„ A path or circuit is simple if it contains no vertex more than once. (simple circuit通常稱為cycle)

Paths in Directed Graphs

„ Same as in undirected graphs, but the

path must go in the direction of the arrows.

Figure 5.

u v

Connectedness

Def. 3:

An undirected graph is connected (連通) iff there is a path between every pair of distinct vertices in the graph.

Def:

Connected component: maximal

connected subgraph. (一個不連通的圖 會有好幾個component)

Example 6

Connectedness

Def:

A cut vertex separates one connected component into several components if it is removed.

Def:

A cut edge separates one connected component into two components if it is removed.

Example 8.

Connectedness in Digraphs

Def. 4:

A directed graph is strongly connected iff there is a directed path from a to b for any two

vertices a and b.

Def. 5:

It is weakly connected iff the underlying

undirected graph (i.e., with edge directions removed) is connected .

„ Note strongly implies weakly but not vice-versa.

Connectedness in Digraphs

Example 9.

„ G is strongly/weakly connected

„ H is only weakly connected

Paths & Isomorphism

„ Note that connectedness, and the existence of a circuit or simple circuit of length k are graph

invariants with respect to isomorphism.

„ Example 12: Determine whether the graphs G and H shown in Figure 6 are isomorphic

Example 12

„ Sol: Both G and H have six vertices and eight edges. Each has four vertices of degree three, and two vertices of degree two.

„ However, H has a simple circuit of length three, namely, v1, v2, v6, v1, whereas G has no simple circuit of length three, as can be determined by inspection (all simple circuits in G gave length at least four). Because the existence of simple

circuit of length three is an isomorphic invariant, G and H are not isomorphic.

Paths & Isomorphism

„ Example 13: Determine whether the graphs G and H shown in Figure 7 are isomorphic

Example 13

„ Both G and H have five vertices and six edges. Each has two vertices of degree three, and three vertices of degree two, and both have a simple circuit of length

three, a simple circuit of length four, and a simple circuit of length five.

„ G and H are isomorphic.

Counting Paths between Vertices

Let A be the adjacency matrix of graph G.

Theorem 2:

The number of paths of length r from v_i to v_j is equal to (A^r)_i,j. (The notation (M)_i,j denotes m_i,j where [m_i,j] = M.)

Example 14.

Exercise: 15, 23, 25, 26

Example 14.

„ How many paths of length four are there from a to d in the simple graph G in Figure 8?

„ Sol: a,b,a,b,d; a,b,a,c,d; a,b,d,b,d;

a,b,d,c,d; a,c,a,b,d; a,c,a,c,d; a,c,d,b,d;

a,c,d,c,d.

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

0 1

1 0

1 0

0 1

1 0

0 1

0 1

1 0

A

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

8 0

0 8

0 8

8 0

0 8

8 0

8 0

0 8

A4

§8.5: Euler & Hamilton Paths

Def. 1:

„ An Euler circuit in a graph G is a simple circuit containing every edge of G.

„ An Euler path in G is a simple path containing every edge of G.

Example 1.

Example 1

Example 2

„ Which of the directed graphs in Figure 4 have an Euler circuit? Of those that do not, which have Euler path?

Example

„ Sol: The graph H2 has Euler circuit, for example, a,g,c,b,g,e,d,f,a.

„ Neither H1 nor H3 has Euler circuit. H3

has an Euler path, namely, c,a,b,c,d,b, but H1 does not.

Useful Theorems

Thm. 1:

A connected multigraph has an Euler circuit iff each vertex has even degree.

Thm. 2:

A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree.

Example 4

„ Which graphs shown in Figure 7 have an Euler path?

Example 4

„ Sol: G1 contains exactly two vertices of odd

degree, namely, b and d. Hence, it has an Euler path that must have b and d as its endpoints.

One such Euler path is d,a,b,c,d,b.

„ Similarly, G2 has exactly two vertices of odd

degree, namely, b and d. So it has an Euler path that must have b and d as endpoints. One such Euler path is b,a,g,f,e,d,c,g,b,c,f,d.

„ G3 has no Euler path because is has six vertices of odd degree.

Hamilton Paths

Def. 2:

„ A Hamilton circuit is a circuit that

traverses each vertex in G exactly once.

„ A Hamilton path is a path that traverses each vertex in G exactly once.

Example 5

„ Which of the simple graphs in Figure 10 have a Hamilton circuit or, if not, a Hamilton Paths?

Example 5

„ Sol: G1 has a Hamilton circuit: a,b,c,d,e,a.

„ There is no Hamilton circuit in G2 (this can be seen by nothing that any circuit containing every vertex must contain the edge {a,b} twice), but G2 does have a Hamilton path, namely, a,b,c,d.

„ G3 has neither a Hamilton circuit nor a Hamilton path, because any path containing all vertices

must contain one of the edges {a,b},{e,f}, and {c,d} more than once.

Useful Theorems

Thm. 3 (Dirac’s Thm.):

If (but not only if) G is connected, simple, has n≥3 vertices, and deg(v)≥n/2 ∀v, then G has a Hamilton circuit.

Exercise: 3, 5, 7, 21, 26, 28, 42, 43.

§8.6: Shortest Path Problems

Def:

Graphs that have a number assigned to each edge are called weighted graphs.

Shortest path Problem:

Determining the path of least sum of the weights between two vertices in a

weighted graph.

Dijkstra’s Algorithm

Figure 4.

Exercise: 3

§8.7: Planar Graphs

Def. 1:

A graph is called planar if it can be drawn in the plane without any edge crossing.

Exercise: 2,3,4

Example 1

„ K₄ is planar.

Example 3

„ K_3,3 is nonplanar.

§8.8: Graph Coloring

Def. 1:

A coloring of a simple graph is the

assignment of a color to each vertex of the graph so that no two adjacent

vertices are assigned the same color.

Def. 2:

The chromatic number of a graph is the least number of colors needed for a

coloring of this graph. (denoted by χ(G))

§8.8: Graph Coloring

Example 2~4: χ(K_n)=n, χ(K_m,n)=2, χ(C_n)=2,3.

Example 3’: If G is a bipartite graph, χ(G)=2.

Theorem 1. (The Four Color Theorem)

The chromatic number of a planar graph is no greater than four.

Exercise: 6, 7

Example 1

„ What are the chromatic number of graphs G and H show in Figure 3?

Example 2

„ What is the chromatic number of K_n ?

„ Sol: Here, the chromatic number of K_n = n.That is, χ(K_n)=n. (Recall that K_n is not planar when n >=5, so this does not contradict the four color Theorem.) A coloring of K₅ using five colors is shown in Figure 5.