A simple graph consists of , a nonempty set of vertices, and , a - - PowerPoint PPT Presentation

SLIDE 1 ✁ ✂✄ ☎ ✆✝ ✆ ✞ ✝ ☎ ✟ ✄ ✝ ✟ ☎ ✆ ✂ ✠ ✡ ☛ ☞✍✌ ✎ ✏ ✑ ✒ ✓ ✔✕ ✖ ✗ ✘ ✙ ✕ ✚ ✔ ✛ ✜ ✢ ✣ ✗ ✤✥ ✦ ✗ ✤ ✧ ★ ✩✪✫ ✪✬ ✭ ✮✯ ✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ❈ ❉ ❊ ❋ ✁ ☎ ✆ ✄ ✝ ✆ ❋

• ☎

❍ ■✍❏ ✂ ❑

A simple graph

▲ ▼ ◆ ❖ P ◗ ❘

consists of

❖

, a nonempty set of vertices, and

◗

, a set of unordered pairs of distinct elements of

❖

called edges.

❑

A multigraph

▲ ▼ ◆ ❖ P ◗ ❘

consists of a set

❖

• f vertices, a set

◗

• f edges,

and a function

❙

from

◗

to

❚ ❚❱❯ P ❲ ❳❨ ❯ P ❲ ❩ ❖ P ❯ ❬ ▼ ❲ ❳

. The edges

❭ ❪

and

❭ ❫

are called multiple or parallel edges if

❙ ◆ ❭ ❪ ❘ ▼ ❙ ◆ ❭ ❫ ❘

.

❑

A pseudograph

▲ ▼ ◆ ❖ P ◗ ❘

consists of a set

❖

• f vertices, a set

◗

• f

edges, and a function

❙

from

◗

to

❚ ❚ ❯ P ❲ ❳❨ ❯ P ❲ ❩ ❖ ❳

. An edge is a loop if

❙ ◆ ❭ ❘ ▼ ❚ ❯ P ❯ ❳ ▼ ❚❱❯ ❳

for some

❯ ❩ ❖

.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ✼ ✁ ☎ ✆ ✄ ✝ ✆ ❋

• ☎

❍ ■✍❏ ✂ ❑

A directed graph

▲ ▼ ◆ ❖ P ◗ ❘

consists of a set

❖

• f vertices and a set of

edges

◗

that are ordered pairs of elements of

❖

.

❑

A directed multigraph

▲ ▼ ◆ ❖ P ◗ ❘

consists of a set

❖

• f vertices, a set

◗

• f edges, and a function

❙

from

◗

to

❚ ◆ ❯ P ❲ ❘ ❨ ❯ P ❲ ❩ ❖ ❳

. The edges

❭ ❪

and

❭ ❫

are multiple edges if

❙ ◆ ❭ ❪ ❘ ▼ ❙ ◆ ❭ ❫ ❘

.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ✻ ❉ ❊ ❋ ✁ ☎ ✆ ✄ ✝ ✆ ❋

• ☎

❍ ■✍❏ ❴ ✆ ☎ ❵ ✁ ❊ ❛ ❜ ❛ ❝❞ ❑

Two vertices

❯

and

❲

in an undirected graph

▲

are called adjacent (or neighbors) in

▲

if

❚ ❯ P ❲ ❳

is an edge of

▲

. If

❭ ▼ ❚❱❯ P ❲ ❳

, the edge

❭

is called incident with the vertices

❯

and

❲

. The edge

❭

is also said to connect

❯

and

❲

. The vertices

❯

and

❲

are called endpoints of the edges

❚ ❯ P ❲ ❳

.

❑

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 the vertex

❲

is denoted by deg(

❲

).

❑

The Handshaking Theorem : Let

▲ ▼ ◆ ❖ P ◗ ❘

be an undirected graph with

❭

• edges. Then

❡ ❭ ▼ ❢ ❣ ❤ ✐

deg

◆ ❲ ❘❦❥ ❑

Theorem : An undirected graph has an even number of vertices of

• dd degree.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ✺

SLIDE 2

• ✁

☎ ✆ ✄ ✝ ✆ ❋

• ☎

❍ ■✍❏ ❴ ✆ ☎ ❵ ✁ ❊ ❛ ❜ ❛ ❝❞ ❑

When

◆ ❯ P ❲ ❘

is an edge of the graph

▲

with directed edges,

❯

is said to be adjacent to

❲

and

❲

is said to be adjacent from

❯

. The vertex

❯

is called the initial vertex of

◆ ❯ P ❲ ❘

, and

❲

is called the terminal or end vertex of

◆ ❯ P ❲ ❘

. The initial vertex and terminal vertex of a loop are the same.

❑

In a graph with directed edges the in-degree of a vertex

❲

, denoted by deg

❧ ◆ ❲ ❘

, is the number of edges with

❲

as their terminal vertex. The

• ut-degree of

❲

, denoted by deg

♠ ◆ ❲ ❘

, is the number of edges with

❲

as their initial vertex.

❑

Theorem: Let

▲ ▼ ◆ ❖ P ◗ ❘

be a graph with directed edges. Then

❢ ❣ ❤ ✐

deg

❧ ◆ ❲ ❘ ▼ ❢ ❣ ❤ ✐

deg

♠ ◆ ❲ ❘ ▼ ❨ ◗ ❨ ❥ ✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ♥ ♦ ❛ ☎ ✆

• ✆

♣ ❊ ✁ ✝ ✁ ❛ ❊ ✂ q q q ❑

A simple graph is

▲

is called bipartite if its vertex

❖

can be partitioned into two disjoint nonempty sets

❖ ❪

and

❖ ❫

such that every edge in the graph connects a vertex in

❖ ❪

and a vertex in

❖ ❫

(so that no edge in

▲

connects either two vertices in

❖ ❪

• r two vertices in

❖ ❫

.

❑

A subgraph of a graph

▲ ▼ ◆ ❖ P ◗ ❘

is a graph

r ▼ ◆ s P t ❘

where

s ✉ ❖

and

t ✉ ◗

.

❑

The union of two simple graphs

▲ ❪ ▼ ◆ ❖ ❪ P ◗ ❪ ❘

and

▲ ❫ ▼ ◆ ❖ ❫ P ◗ ❫ ❘

is the simple graph with vertex set

❖ ❪ ✈ ❖ ❫

and edge set

◗ ❪ ✈ ◗ ❫

. The union

• f

▲ ❪

and

▲ ❫

is denoted by

▲ ❪ ✈ ▲ ❫

.

❑

The simple graphs

▲ ❪ ▼ ◆ ❖ ❪ P ◗ ❪ ❘

and

▲ ❫ ▼ ◆ ❖ ❫ P ◗ ❫ ❘

are isomorphic if there is a one-to-one and onto function

❙

from

❖ ❪

to

❖ ❫

with the property that

✇

and

①

are adjacent in

▲ ❪

if and only if

❙ ◆ ✇ ❘

and

❙ ◆ ① ❘

are adjacent in

▲ ❫

, for all

✇

and

①

in

❖ ❪

. Such a function

❙

is called an isomorphism.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ② ③ ❛ ❊ ❊ ✆ ✄ ✝ ✁ ④ ✁ ✝ ❞ ⑤ ❑

A path of lengh

⑥

from

❯

to

❲

, where

⑥

is a positive integer, in an undirected graph is a sequence of edges

❭ ❪ P ❭ ❫ P ❥ ❥ ❥ P ❭ ⑦

• f the graph

such that

❙ ◆ ❭ ❪ ❘ ▼ ❚❱⑧ ⑨ P ⑧ ❪ ❳ P ❙ ◆ ❭ ❫ ❘ ▼ ❚❱⑧ ❪ P ⑧ ❫ ❳ P ❥ ❥ ❥ ❙ ◆ ❭ ⑦ ❘ ▼ ❚ ⑧ ⑦ ❧ ❪ P ⑧ ⑦ ❳

, where

⑧ ⑨ ▼ ❯

and

⑧ ⑦ ▼ ❲

. When the graph is simple, we denote this path by its vertex sequence

⑧ ⑨ P ⑧ ❪ P ❥ ❥ ❥ P ⑧ ⑦

. The path is a circuit if it begins and ends at the same vertex. The path or circuit is said to pass through or traverse the vertices

⑧ ❪ P ⑧ ❫ P ❥ ❥ ❥ P ⑧ ⑦ ❧ ❪

. A path or circuit is simple if it does not contain the same edge more than once.

❑

A path of lengh

⑥

from

❯

to

❲

in a directed multigraph, where

⑥

is a positive integer, is a sequence of edges

❭ ❪ P ❭ ❫ P ❥ ❥ ❥ P ❭ ⑦

• f the graph such that

❙ ◆ ❭ ❪ ❘ ▼ ◆ ⑧ ⑨ P ⑧ ❪ ❘ P ❙ ◆ ❭ ❫ ❘ ▼ ◆ ⑧ ❪ P ⑧ ❫ ❘ P ❥ ❥ ❥ ❙ ◆ ❭ ⑦ ❘ ▼ ◆ ⑧ ⑦ ❧ ❪ P ⑧ ⑦ ❘

, where

⑧ ⑨ ▼ ❯

and

⑧ ⑦ ▼ ❲

. When there are no multiple edges in the graph, we denote this path by its vertex sequence

⑧ ⑨ P ⑧ ❪ P ❥ ❥ ❥ P ⑧ ⑦

. The path is a circuit or cycle if it begins and ends at the same vertex. A path or circuit is simple if it does not contain the same edge more than once.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ⑩ ③ ❛ ❊ ❊ ✆ ✄ ✝ ✁ ④ ✁ ✝ ❞ ❶ ❑

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.

❑

A directed graph is strongly connected if there is a path from

✇

to

①

and from

①

to

✇

whenever

✇

and

①

are vertices in the graph.

❑

A directed graph is weakly connected if there is a path between any two vertices in the underlying undirected graph.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ❃

SLIDE 3 ❷ ✟ ❜ ✆ ☎ ③ ✁ ☎ ✄ ✟ ✁ ✝ ✂ ❑

An Euler circuit in a graph

▲

is a simple circuit containing every edge of

▲

.

❑

Theorem: A connected multigraph has an Euler circuit if and only if each

• f its vertices has even degree.

✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ❸