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I118 Graphs and Automata Takako Nemoto http://www.jaist.ac.jp/˜t-nemoto/teaching/2013-1-1.html April 30

I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

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Page 1: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

I118 Graphs and Automata

Takako Nemoto

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

April 30

Page 2: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

0. \!

1. 0sN|,

(a) 0sNdj

2. O_kHs&0iUN~Q

(a) ds;<k9^sdj

(b) G;)dj

3. Z

(a) ZN-A

(b) 4hZ

(c) Z5w

(d) },dj

(e) G."kRdj

Page 3: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

1. 0sN|,

• "k0iU GN9YFNUr^`D8?.;,"kH-, Gr

*$i<&0iU (Eulerian graph)H$$,=N.;r*$i<.;

(Eulerian trail)H$&.

• j} (Euler, 1736/): "k0iU G,*$i<&0iUG"k,W=,

roO GN:@N!t,9YFvtG"k3HG"k.

• "k0iU G,,9YFN:@rlY@1LkD)r}DH-, Gr

O_kHs&0iU (Hamiltonian graph)H$$,=ND)r

O_kHsD) (Hamiltonian cycle)H$&.

• j}: 0iU GN:@ v N!tr deg(v)H9k.

G, n(≥ 3)DN@rbDH-,Y\7F$J$$UN 2@ v H w G

deg(v) + deg(w) ≥ n,.)9kH-, GOO_kHs&0iUG"k.

1

7

2

3

6

4

5

1

2 3

4

*$i<&0iUNc

c: 3DJeN:@r}D040iUOO_kHs&0iU

Page 4: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

1a. 0sNdj

:@N8g V ,sDN_$KBJ8g V1, V2 K,dG-,9YFNU, V1

N@H V2 N@rkVst0iUNfG,9YFN V1 N@H V2 N@NH

(u, v)KD$F uH v rkVU,"kbNr04st0iUH$&.

1. eNjAG, V1 = {1, 2}, V2 = {3, 4}H9k04st0iUrA1.

2. eGA$?0iUO*$i<&0iU+)

3. eGA$?0iUOO_kHs&0iU+)

4. |V1| = j, |V2| = k N04st0iUKO?\NU,"k+)

5. 04st0iU,O_kHs&0iUKJk?aN,W=,roO?+)

Page 5: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2a. ds;<k9^sdj

“ds;<k9^s,hail?$/D+N9rsCFO/N9KakH-,

G;NgW\0w%KJkP)raak"k4j:`O8_9k+)”

3NdjO 0iUrH&H!Nh&K=;k

F,dhr H7 3lir:@KbD040iUrq/

FUK =liK\39k9H9HNVNw%rdjvFk

3N0iUG FUKdj6il?MNgWM,G.HJkO_kHsD

)raak"k4j:`O8_9k+)

3Nh&K 0iUNFUKtM,djvFi

lF$k"k0iUrE_D-0iU

H$&

9YFNO_kHsD)KD$FW;9k}!O

"^j=B*GJ$

BOI$ ?`0~VN "k4j:`ONilF

$J$ $q

mU ?(il?0iU,O_kHs&0iU+I&+r4Yk"k4j:

`O 04

Page 6: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2a. ds;<k9^sdj

“ds;<k9^s,hail?$/D+N9rsCFO/N9KakH-,

G;NgW\0w%KJkP)raak"k4j:`O8_9k+)”

3NdjO,0iUrH&H!Nh&K=;k.

• F,dhr v1, ..., vn H7,3lir:@KbD040iUrq/

• FUK,=liK\39k9H9HNVNw%rdjvFk.

• 3N0iUG,FUKdj6il?MNgWM,G.HJkO_kHsD

)raak"k4j:`O8_9k+)

A

B

3

E

2

C

6

D

13

4 5 4 4

2

3Nh&K 0iUNFUKtM,djvFi

lF$k"k0iUrE_D-0iU

H$&

9YFNO_kHsD)KD$FW;9k}!O

"^j=B*GJ$

BOI$ ?`0~VN "k4j:`ONilF

$J$ $q

mU ?(il?0iU,O_kHs&0iU+I&+r4Yk"k4j:

`O 04

Page 7: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2a. ds;<k9^sdj

“ds;<k9^s,hail?$/D+N9rsCFO/N9KakH-,

G;NgW\0w%KJkP)raak"k4j:`O8_9k+)”

3NdjO,0iUrH&H!Nh&K=;k.

• F,dhr v1, ..., vn H7,3lir:@KbD040iUrq/

• FUK,=liK\39k9H9HNVNw%rdjvFk.

• 3N0iUG,FUKdj6il?MNgWM,G.HJkO_kHsD

)raak"k4j:`O8_9k+)

A

B

3

E

2

C

6

D

13

4 5 4 4

2

3Nh&K, 0iUNFUKtM,djvFi

lF$k"k0iUrE_D-0iU (weighted

graph)H$&.

9YFNO_kHsD)KD$FW;9k}!O

"^j=B*GJ$

BOI$ ?`0~VN "k4j:`ONilF

$J$ $q

mU ?(il?0iU,O_kHs&0iU+I&+r4Yk"k4j:

`O 04

Page 8: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2a. ds;<k9^sdj

“ds;<k9^s,hail?$/D+N9rsCFO/N9KakH-,

G;NgW\0w%KJkP)raak"k4j:`O8_9k+)”

3NdjO,0iUrH&H!Nh&K=;k.

• F,dhr v1, ..., vn H7,3lir:@KbD040iUrq/

• FUK,=liK\39k9H9HNVNw%rdjvFk.

• 3N0iUG,FUKdj6il?MNgWM,G.HJkO_kHsD

)raak"k4j:`O8_9k+)

A

B

3

E

2

C

6

D

13

4 5 4 4

2

3Nh&K, 0iUNFUKtM,djvFi

lF$k"k0iUrE_D-0iU (weighted

graph)H$&.

9YFNO_kHsD)KD$FW;9k}!O,

"^j=B*GJ$.

BOI$ (?`0~VN)"k4j:`ONilF

$J$ (NP$q).

mU ?(il?0iU,O_kHs&0iU+I&+r4Yk"k4j:

`O 04

Page 9: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2a. ds;<k9^sdj

“ds;<k9^s,hail?$/D+N9rsCFO/N9KakH-,

G;NgW\0w%KJkP)raak"k4j:`O8_9k+)”

3NdjO,0iUrH&H!Nh&K=;k.

• F,dhr v1, ..., vn H7,3lir:@KbD040iUrq/

• FUK,=liK\39k9H9HNVNw%rdjvFk.

• 3N0iUG,FUKdj6il?MNgWM,G.HJkO_kHsD

)raak"k4j:`O8_9k+)

A

B

3

E

2

C

6

D

13

4 5 4 4

2

3Nh&K, 0iUNFUKtM,djvFi

lF$k"k0iUrE_D-0iU (weighted

graph)H$&.

9YFNO_kHsD)KD$FW;9k}!O,

"^j=B*GJ$.

BOI$ (?`0~VN)"k4j:`ONilF

$J$ (NP$q).

mU: ?(il?0iU,O_kHs&0iU+I&+r4Yk"k4j:

`O NP04.

Page 10: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ +i XNG;P)Naa}

KJWiYk r=k

!K KY\7?@K>NiYkr +iNw% H{K=i

lF$kiYkHGMN.5$bNH9k

>iYk,=ilF$kbNNf+iG.NM,=ilF$k@ r

*S JWiYk r>iYkNMH9k

H7F Kak

NJWiYk,h^C?i =l, H NVNG;P)G"k

BO3Nj!O4;NP)FbKbQ$ilF$k

Page 11: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ u+i v XNG;P)Naa}

1. uKJWiYk l(u) = 0r=k.

!K KY\7?@K>NiYkr +iNw% H{K=i

lF$kiYkHGMN.5$bNH9k

>iYk,=ilF$kbNNf+iG.NM,=ilF$k@ r

*S JWiYk r>iYkNMH9k

H7F Kak

NJWiYk,h^C?i =l, H NVNG;P)G"k

BO3Nj!O4;NP)FbKbQ$ilF$k

Page 12: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ u+i v XNG;P)Naa}

1. uKJWiYk l(u) = 0r=k.

2. !K uKY\7?@K>NiYkr l(u) + (u+iNw%)H{K=ilF$kiYkHGMN.5$bNH9k.

>iYk,=ilF$kbNNf+iG.NM,=ilF$k@ r

*S JWiYk r>iYkNMH9k

H7F Kak

NJWiYk,h^C?i =l, H NVNG;P)G"k

BO3Nj!O4;NP)FbKbQ$ilF$k

Page 13: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ u+i v XNG;P)Naa}

1. uKJWiYk l(u) = 0r=k.

2. !K uKY\7?@K>NiYkr l(u) + (u+iNw%)H{K=ilF$kiYkHGMN.5$bNH9k.

3. >iYk,=ilF$kbNNf+iG.NM,=ilF$k@ w r

*S,JWiYk l(w)r>iYkNMH9k.

H7F Kak

NJWiYk,h^C?i =l, H NVNG;P)G"k

BO3Nj!O4;NP)FbKbQ$ilF$k

Page 14: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ u+i v XNG;P)Naa}

1. uKJWiYk l(u) = 0r=k.

2. !K uKY\7?@K>NiYkr l(u) + (u+iNw%)H{K=ilF$kiYkHGMN.5$bNH9k.

3. >iYk,=ilF$kbNNf+iG.NM,=ilF$k@ w r

*S,JWiYk l(w)r>iYkNMH9k.

4. u = w H7F 2Kak.

NJWiYk,h^C?i =l, H NVNG;P)G"k

BO3Nj!O4;NP)FbKbQ$ilF$k

Page 15: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ u+i v XNG;P)Naa}

1. uKJWiYk l(u) = 0r=k.

2. !K uKY\7?@K>NiYkr l(u) + (u+iNw%)H{K=ilF$kiYkHGMN.5$bNH9k.

3. >iYk,=ilF$kbNNf+iG.NM,=ilF$k@ w r

*S,JWiYk l(w)r>iYkNMH9k.

4. u = w H7F 2Kak.

5. v NJWiYk,h^C?i,=l, uH v NVNG;P)G"k.

BO3Nj!O4;NP)FbKbQ$ilF$k

Page 16: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

2b. G;P)dj

A

B

3

E

2

C

6

D

13

4 5 4 4

2

8NFUKdjvFil?tMrF@VNw%H

9k. $UN 2@NG;P)raak"k4j:`O)

O@ u+i v XNG;P)Naa}

1. uKJWiYk l(u) = 0r=k.

2. !K uKY\7?@K>NiYkr l(u) + (u+iNw%)H{K=ilF$kiYkHGMN.5$bNH9k.

3. >iYk,=ilF$kbNNf+iG.NM,=ilF$k@ w r

*S,JWiYk l(w)r>iYkNMH9k.

4. u = w H7F 2Kak.

5. v NJWiYk,h^C?i,=l, uH v NVNG;P)G"k.

BO3Nj!O4;NP)FbKbQ$ilF$k.

Page 17: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3. Z

jA D)r^^J$0iUrS (forest),"kJSrZ (tree)H$&.

c

Page 18: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3. Z

jA D)r^^J$0iUrS (forest),"kJSrZ (tree)H$&.

c 1

2 3

4 5 6 7 8

9

1

2 5

3 4 6 7 8

9

1

23

4

5

6 7

8

10 11

9

1

5

2

6

4

3

Page 19: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3a. ZN-A

j} 0iU T ,:@r nDbDH9k. !N?jO1MG"k.

1. T OZG"k.

2. T KD)OJ/,U, n− 1\"k.

3. T O"kG,U, n− 1\"k.

4. T O"kG"j,9YFNUO6G"k.

5. T N$UN 2@rkV;OAg&I 1\G"k.

6. T KD)OJ$,,77$UrI&U1C(Fb 1D@1D),G-k.

O S , DN:@H DN.,rbDH9k 3NH- KOU,

\"k

Page 20: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3a. ZN-A

j} 0iU T ,:@r nDbDH9k. !N?jO1MG"k.

1. T OZG"k.

2. T KD)OJ/,U, n− 1\"k.

3. T O"kG,U, n− 1\"k.

4. T O"kG"j,9YFNUO6G"k.

5. T N$UN 2@rkV;OAg&I 1\G"k.

6. T KD)OJ$,,77$UrI&U1C(Fb 1D@1D),G-k.

O S G, nDN:@H k DN.,rbDH9k. 3NH- GKOU,

n− k \"k.

Page 21: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3b. 4hZ

jA "k0iU G+iD)r*S,=N 1\r|n9k3HrD),J

/Jk^G+jV7F@il?0iUOeNj}hjZHJk. 3&7F

@il?Zr4hZ (spanning tree)H$&.

}, <N0iUN4hZr 3D"2h.

1

2 3

4

Page 22: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3c. 5wZ

<Nh&K,X=$KJCF$kZNGetN 1@rk<H (root)H$&.

a

b c

d e f g h

i

F:@KJ<5l?psN57}:

}%h5w(breadth first search)

<5%h5w(depth first search)

1

2 3

4 5 6 7 8

9

1

2 5

3 4 6 7 8

9

Page 23: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

3d. },

• !NZK}%h5w,<5%h5wG,lkgK:@KiYkr=l.

a

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Page 27: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l6.pdf2013/01/01  · 1a. 0sNdj:@N8g V,sDN_$KBJ8g V1, V2 K,dG-, 9YFNU, V1 N@H V2 N@rkVst0iUNfG, 9YFN V1 N@H V2 N@NH (u,v)KD$F uH

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