I118 Graphs and Automata

Takako Nemoto

http://www.jaist.ac.jp/˜t-nemoto/teaching/2013-1-1.html

April 23

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1. 0sN|,

Xt f : A → B, Ci ⊂ AKD$F,

• C0 ∩ C1 ⊂ C0, C0 ∩ C1 ⊂ C1 r(7J5$.

• f(C0 ∩ C1) ⊂ f(C0) ∩ f(C1)r(7J5$.

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[V ]2 = {{x, y} : x ∈ V ∧ y ∈ V }.

c

c

2. 5~0iU

jA 5~0iU (undirected graph) (V,E)HO,8g V H E ⊂ [V ]2 G?(ilk. 33G

[V ]2 = {{x, y} : x ∈ V ∧ y ∈ V }.

c V = {1, 2, 3, 4}, E = {{1, 2}, {2, 3}, {3, 4}, {1, 3}, {2, 4}, {1, 4}}

1

2

3 4

c V = {1, 2, 3, 4}, E = {{1}, {2, 3}}

1

2 3

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2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

NWGrU

NH-

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2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

NH-

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mU 8N N!tO

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2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

@ KP7F rk<W

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!t N@rI)@

!t N@r<@ H$&

mU 8N N!tO

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk

2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

• @ aKP7F aark<W (loop)

:@ ,\37FkUNtr!t

!t N@rI)@

!t N@r<@ H$&

mU 8N N!tO

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk

2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

• @ aKP7F aark<W (loop)

• :@ v ,\37FkUNtr!t (degree)

!t N@rI)@

!t N@r<@ H$&

mU 8N N!tO

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk

2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

• @ aKP7F aark<W (loop)

• :@ v ,\37FkUNtr!t (degree)

• !t 0N@rI)@ (isolated point)

!t N@r<@ H$&

mU 8N N!tO

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk

2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

• @ aKP7F aark<W (loop)

• :@ v ,\37FkUNtr!t (degree)

• !t 0N@rI)@ (isolated point)

• !t 1N@r<@ (end-vertix) H$&.

mU 8N N!tO

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk

2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

• @ aKP7F aark<W (loop)

• :@ v ,\37FkUNtr!t (degree)

• !t 0N@rI)@ (isolated point)

• !t 1N@r<@ (end-vertix) H$&.

mU a 8N aN!tO 2.

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk

2a. $m$mJQl

5~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)

• E NWGrU (edge)

• {a, b} ∈ E NH-

• aH bOY\ (adjacent)7F$k

• aH bOU ab(= {a, b})K\3 (incident)7F$k

• @ aKP7F aark<W (loop)

• :@ v ,\37FkUNtr!t (degree)

• !t 0N@rI)@ (isolated point)

• !t 1N@r<@ (end-vertix) H$&.

mU a 8N aN!tO 2.

.jdj $UN5~0iUN9YFN:@N!trgW9kHvtKJk.

2b. b;H"k

0iU G = (V,E)KP7F

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$&

$UN @NVKb;,"k5~0iUO"k G"k

c "kJ5~0iU "kGJ$5~0iU

dj 5~0iU G r +i XNb;,"k

H$&X8H9kH O1MX8G"k

jA eN KD$F N5r N., H$&

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2b. b;H"k

0iU G = (V,E)KP7F

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$&

• $UN 2@NVKb;,"k5~0iUO"k (connected)G"k.

c "kJ5~0iU "kGJ$5~0iU

dj 5~0iU G r +i XNb;,"k

H$&X8H9kH O1MX8G"k

jA eN KD$F N5r N., H$&

}, eN N.,O)

2b. b;H"k

0iU G = (V,E)KP7F

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$&

• $UN 2@NVKb;,"k5~0iUO"k (connected)G"k.

c "kJ5~0iU G1

1

2 3

4

"kGJ$5~0iU G2

1

2 3

4

dj 5~0iU G r +i XNb;,"k

H$&X8H9kH O1MX8G"k

jA eN KD$F N5r N., H$&

}, eN N.,O)

2b. b;H"k

0iU G = (V,E)KP7F

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$&

• $UN 2@NVKb;,"k5~0iUO"k (connected)G"k.

c "kJ5~0iU G1

1

2 3

4

"kGJ$5~0iU G2

1

2 3

4

dj 5~0iU G = (V,E)G a1 ∼ a2 r “a1 +i a2 XNb;,"k”

H$&X8H9kH ∼O1MX8G"k.

jA eN ∼KD$F, G/ ∼N5r GN., (component)H$&.

}, eN N.,O)

2b. b;H"k

0iU G = (V,E)KP7F

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$&

• $UN 2@NVKb;,"k5~0iUO"k (connected)G"k.

c "kJ5~0iU G1

1

2 3

4

"kGJ$5~0iU G2

1

2 3

4

dj 5~0iU G = (V,E)G a1 ∼ a2 r “a1 +i a2 XNb;,"k”

H$&X8H9kH ∼O1MX8G"k.

jA eN ∼KD$F, G/ ∼N5r GN., (component)H$&.

}, eN G1, G2 N.,O)

2c. t,0iU

jA 5~0iU G1 = (V1, E2), G2 = (V2, E2)KD$F V1 ⊂ V2,

E1 ⊂ E2 NH-, G1 O G2 Nt,0iU (subgraph)G"kH$&.

}, !N0iU GNt,0iUO)

a

b c

jA 0iU GHUN8g F KD$F G+i F K^^lkUrhj|$?0iUr G− F H9k.

:@N8g S KD$F, G+i S K^^lk:@H=lK\39kU9YFrhj|$?0iUr G− S H9k.

}, eN G, F = {ab}, S = {c},KD$F G− F,G− S O)

jA 5~0iU NB r

Gjak

2c. t,0iU

jA 5~0iU G1 = (V1, E2), G2 = (V2, E2)KD$F V1 ⊂ V2,

E1 ⊂ E2 NH-, G1 O G2 Nt,0iU (subgraph)G"kH$&.

}, !N0iU GNt,0iUO)

a

b c

jA 0iU GHUN8g F KD$F G+i F K^^lkUrhj|$?0iUr G− F H9k.

:@N8g S KD$F, G+i S K^^lk:@H=lK\39kU9YFrhj|$?0iUr G− S H9k.

}, eN G, F = {ab}, S = {c},KD$F G− F,G− S O)

jA 5~0iU G1 = (V1, E2), G2 = (V2, E2)NB (union) G1 ∪G2 r

(V1 ∪ V2, E1 ∪ E2)Gjak.

2d. 1?

jA 5~0iU G1 = (V1, E1), G2 = (V2, E2)KD$F f : V1 → V2 ,

41MG$UN:@ v1, v2 ∈ V1 KD$F<Nror~?9H-, G1 H

G2 O1? (isomorphic), f r1?L| (isomorphism)H$&.

v1v2 ∈ E1 ↔ f(v1)f(v2) ∈ E2.

c <NsDN0iUO1?G"k.

1

2

3 4

5

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8

}, !N0iUN1?L|O)

2d. 1?

jA 5~0iU G1 = (V1, E1), G2 = (V2, E2)KD$F f : V1 → V2 ,

41MG$UN:@ v1, v2 ∈ V1 KD$F<Nror~?9H-, G1 H

G2 O1? (isomorphic), f r1?L| (isomorphism)H$&.

v1v2 ∈ E1 ↔ f(v1)f(v2) ∈ E2.

c <NsDN0iUO1?G"k.

1

2

3 4

5

6 7

8

}, !N0iUN1?L|O)

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4

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3. -~0iU

|,

-~0iU (directed graph)HO8g V H E ⊂ V × V G?(ilk.

jA -~0iU G = (V,E)KP7F,pC0iU (underlying graph)

G′ = (V ′, E′)HO5~0iUG!Nror~?9bNG"k.

V = V ′ +D (a, b) ∈ E ↔ ab ∈ E′

c -~0iUHpC0iU

1

2 3

4

5 6

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

NWGrU ^?OL H$&

NH-

H OY\ 7F$k

H OU K\3 7F$k

@ KP7F rk<W H$&

:@Ns G$UN KD$F HJk

bNr Nb; H$& Hq/

pC0iU,"kJ-~0iUO"k G"k

$UN @NVKb;,"kH-/"k G"k

}, "k@,/"kGJ$-~0iUrq1

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

• E NWGrU (edge),^?OL (arc)H$&.

NH-

H OY\ 7F$k

H OU K\3 7F$k

@ KP7F rk<W H$&

:@Ns G$UN KD$F HJk

bNr Nb; H$& Hq/

pC0iU,"kJ-~0iUO"k G"k

$UN @NVKb;,"kH-/"k G"k

}, "k@,/"kGJ$-~0iUrq1

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

• E NWGrU (edge),^?OL (arc)H$&.

• (a, b) ∈ E NH-

• aH bOY\ (adjacent)7F$k.

• aH bOU ab(= (a, b))K\3 (incident)7F$k.

@ KP7F rk<W H$&

:@Ns G$UN KD$F HJk

bNr Nb; H$& Hq/

pC0iU,"kJ-~0iUO"k G"k

$UN @NVKb;,"kH-/"k G"k

}, "k@,/"kGJ$-~0iUrq1

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

• E NWGrU (edge),^?OL (arc)H$&.

• (a, b) ∈ E NH-

• aH bOY\ (adjacent)7F$k.

• aH bOU ab(= (a, b))K\3 (incident)7F$k.

• @ aKP7F aark<W (loop)H$&.

:@Ns G$UN KD$F HJk

bNr Nb; H$& Hq/

pC0iU,"kJ-~0iUO"k G"k

$UN @NVKb;,"kH-/"k G"k

}, "k@,/"kGJ$-~0iUrq1

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

• E NWGrU (edge),^?OL (arc)H$&.

• (a, b) ∈ E NH-

• aH bOY\ (adjacent)7F$k.

• aH bOU ab(= (a, b))K\3 (incident)7F$k.

• @ aKP7F aark<W (loop)H$&.

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$& (a0 → a1 → · · · → an Hq/).

• pC0iU,"kJ-~0iUO"k (connected)G"k.

$UN @NVKb;,"kH-/"k G"k

}, "k@,/"kGJ$-~0iUrq1

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

• E NWGrU (edge),^?OL (arc)H$&.

• (a, b) ∈ E NH-

• aH bOY\ (adjacent)7F$k.

• aH bOU ab(= (a, b))K\3 (incident)7F$k.

• @ aKP7F aark<W (loop)H$&.

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$& (a0 → a1 → · · · → an Hq/).

• pC0iU,"kJ-~0iUO"k (connected)G"k.

• $UN 2@NVKb;,"kH-/"k (strongly connected)G"k.

}, "k@,/"kGJ$-~0iUrq1

3a. $m$mJQl

-~0iU G = (V,E)KP7F

• V NWGr:@ (vertix),^?O@,N<I (node)H$&.

• E NWGrU (edge),^?OL (arc)H$&.

• (a, b) ∈ E NH-

• aH bOY\ (adjacent)7F$k.

• aH bOU ab(= (a, b))K\3 (incident)7F$k.

• @ aKP7F aark<W (loop)H$&.

• :@Ns w = a0, a1, ...., an G$UN i < nKD$F aiai+1 ∈ E HJkbNrGNb; (walk)H$& (a0 → a1 → · · · → an Hq/).

• pC0iU,"kJ-~0iUO"k (connected)G"k.

• $UN 2@NVKb;,"kH-/"k (strongly connected)G"k.

}, "k@,/"kGJ$-~0iUrq1.

3b. 1?

jA -~0iU G1 = (V1, V2), G2 = (V2, E2)KD$F f : V1 → V2 ,

41MG$UN:@ v1, v2 ∈ V1 KD$F<Nror~?9H-, f rG1 H G2 O1? (isomorphic), f r1?L| (isomorphism)H$&.

v1v2 ∈ E1 ↔ f(v1)f(v2) ∈ E2.

c <NsDN0iUO1?G"k

}, !NsDN0iUO1?+)

3b. 1?

jA -~0iU G1 = (V1, V2), G2 = (V2, E2)KD$F f : V1 → V2 ,

41MG$UN:@ v1, v2 ∈ V1 KD$F<Nror~?9H-, f rG1 H G2 O1? (isomorphic), f r1?L| (isomorphism)H$&.

v1v2 ∈ E1 ↔ f(v1)f(v2) ∈ E2.

c <NsDN0iUO1?G"k.

1

2 3

4

5 6

}, !NsDN0iUO1?+)

3b. 1?

jA -~0iU G1 = (V1, V2), G2 = (V2, E2)KD$F f : V1 → V2 ,

41MG$UN:@ v1, v2 ∈ V1 KD$F<Nror~?9H-, f rG1 H G2 O1? (isomorphic), f r1?L| (isomorphism)H$&.

v1v2 ∈ E1 ↔ f(v1)f(v2) ∈ E2.

c <NsDN0iUO1?G"k.

1

2 3

4

5 6

}, !NsDN0iUO1?+)

1

2 3

4

5 6