Discrete Mathematics

Chapter-8

Graphs

感謝 大葉大學 資訊工程系 黃鈴玲老師 提供

§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

V5

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} }

Def 2. Multigraph

simple graph + “ 兩點間允許多條邊” multiedge

eg.

V1

V2

V3 V4

V7

V5

V6

Def 3. pseudo graph :

simple graph + multiedge + loop

( loop 即 )

eg.

Table 1. Graph Terminology

Type Edges Multiple Edges

Loops

(simple) graph Undirected表示成 {u,v}

Multigraph

Pseudo graph

Directed graph Directed

表示成 (u,v)

Directed multigraph

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 :

ab

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

Vv

nv 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

Vv

v 32)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

vV

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

(in-degree)

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

Example 3.

a b

c

e df

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

end

deg-(a)=2, deg+(a)=4deg-(b)=2, deg+(b)=1 : : deg-(f)=0, deg+(f)=0

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

Vv Vv

Evv )(deg)(deg

pf :每個 edge 貢獻一個 out–degree 給 a

一個 in–degree 給 ba b

Def : The complete graph on n vertices, denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices.

Example 4.

K1 K2K3 K4

Note : | E | =n

2

Example 5. The cycle Cn, n 3, consists of n ≧vertices v1,v2,…,vn and edges

{v1,v2}, {v2,v3},…,{vn-1,vn},{vn,v1}.

C5 C6

| E | = n

Example 6. Wn : (wheel), Cn 中加一點連至其餘 n 點 (n 3)≧

W5W6

| 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 (km,n)

K2,3K3,3

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

cd

e

a

e b

c

K5

subgraph of K5

Def 7. The union of two simple graph G1=(V1,E1) and G2=(V2,E2) is the simple graph G1 G∪ 2=(V1 V∪ 2,E1 E∪ 2) 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 , vV.

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

ed

c

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,…,vn}. ( 順序沒關系 )

A matrix A is called the adjacency matrix of

G if A=[aij]nxn , where

0011

0011

1101

1110

1AExample 3.

a

c

b

d

undirectedgraph 的連通矩陣必“ 對稱”

0111

1001

1001

1110

2A

a b c da

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=[aij] is the adjacency matrix for the directed graph, then

0212

2110

1103

2030

A

a b c da

b

c

d

aij =

1 , if

0 , otherwise

故矩陣未必對稱

a b

cd

vi vj

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,bV1

f is called an isomorphism.

※Isomorphism of Graphs

u1

u3

u2

u4

G

v1

v3

v2

v4

H

G is isomorphicto 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

Example 10.

Determine whether G and H are isomorphic.

HG

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 Hu1 u2

u4 u3

u5

u6

v1

v2

v3

v4

v5v6

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

§8.4: ConnectivityDef. 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, …, xn=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.

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

ConnectednessDef:

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.

Example 9.

Connectedness in DigraphsDef. 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.

Example 9.

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.

Example 13.

Counting Paths between Vertices

Let A be the adjacency matrix of graph G.

Theorem 2: The number of paths of length r from vi to vj is equal to (Ar)i,j. (The notation (M)i,j denotes mi,j where [mi,j] = M.)

Example 14.

Exercise: 15, 23, 25, 26

§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.

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.

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.

Useful Theorems

Thm. 3 (Dirac’s Thm.):If (but not only if) G is connected, simple, has n3 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 ProblemsDef:

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 GraphsDef. 1:

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

Example 1: K4 is planar.

Example 3: K3,3 is nonplanar.

Exercise: 2,3,4

§8.8: Graph ColoringDef. 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 (G))

§8.8: Graph ColoringExample 1

Example 2~4: (Kn)=n, (Km,n)=2, (Cn)=2,3.

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

Theorem 1. (The Four Color Theorem)The chromatic number of a planar graph isno greater than four.

Exercise: 6, 7