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I118 Graphs and Automata Takako Nemoto http://www.jaist.ac.jp/˜t-nemoto/teaching/2013-1-1.html April 26
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I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

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Page 1: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

I118 Graphs and Automata

Takako Nemoto

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

April 26

Page 2: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

0. \!

1. EW"m

2. b;HD)

(a) "k-

(b) D)Hst0iU

3. *$i<0iU

(a) *$i<0iUN-A

4. O_kHs0iU

(a) O_kHs0iUN-A

Page 3: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

1. EW"m

EW"m

• fVn3O 5n 7|.

• *U#9"o<KD$F

Page 4: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2. b;HD)

#e7Pi/O5~0iUr7&.

|, 0iU G = (V,E)KP7F

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

• $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

jA b;K^^lkUN\tr b;N95H$&

9YFNU,[Jkb;r.;

N&A H J0O

IN @b[Jk.;r;

N.;d;rD8F$k

CKD8?;N3HrD) H$&

c &eN0iUG

Ob;

O.;

OD8?.;

OD)

bD)

Page 5: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2. b;HD)

#e7Pi/O5~0iUr7&.

|, 0iU G = (V,E)KP7F

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

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

jA b;K^^lkUN\tr,b;N95H$&

• 9YFNU,[Jkb;r.; (trail)

• a0, ..., an−1, an N&A, a0 H an J0OIN 2@b[Jk.;r; (path)

• a0 = a1 N.;d;rD8F$k• CKD8?;N3HrD) (cycle)H$&

v w

x

yz

c &eN0iUG

Ob;

O.;

OD8?.;

OD)

bD)

Page 6: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2. b;HD)

#e7Pi/O5~0iUr7&.

|, 0iU G = (V,E)KP7F

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

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

jA b;K^^lkUN\tr,b;N95H$&

• 9YFNU,[Jkb;r.; (trail)

• a0, ..., an−1, an N&A, a0 H an J0OIN 2@b[Jk.;r; (path)

• a0 = a1 N.;d;rD8F$k• CKD8?;N3HrD) (cycle)H$&

v w

x

yz

c &eN0iUG

• v→w→x→w→y→xOb;• v→w→x→y→z→z→xO.;• v→w→x→y→z→x→v OD8?.;

• v→w→x→y→v OD)• z→z bD)

Page 7: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2a. "k-

|,

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

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

H$&X8H9kH ∼O1MX8G"k.

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

mU eNjAN “b;”r “;”K7Fb1M.

c IN @bUGkPlF$k040iU O

?@lDN.,r}D

}, DN:@r}D040iUO \NUr}D

j} , DN@H DN.,rbDk<WNJ$0iUNH-

NUN\t O!N0r~?9

O DN@r}D0iUK \JeNU,"lP"k

Page 8: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2a. "k-

|,

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

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

H$&X8H9kH ∼O1MX8G"k.

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

mU eNjAN “b;”r “;”K7Fb1M.

c IN 2@bUGkPlF$k040iU (complete graph)O

?@lDN.,r}D.

}, DN:@r}D040iUO \NUr}D

j} , DN@H DN.,rbDk<WNJ$0iUNH-

NUN\t O!N0r~?9

O DN@r}D0iUK \JeNU,"lP"k

Page 9: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2a. "k-

|,

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

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

H$&X8H9kH ∼O1MX8G"k.

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

mU eNjAN “b;”r “;”K7Fb1M.

c IN 2@bUGkPlF$k040iU (complete graph)O

?@lDN.,r}D.

}, nDN:@r}D040iUO 1

2n(n− 1)\NUr}D.

j} , DN@H DN.,rbDk<WNJ$0iUNH-

NUN\t O!N0r~?9

O DN@r}D0iUK \JeNU,"lP"k

Page 10: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

2a. "k-

|,

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

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

H$&X8H9kH ∼O1MX8G"k.

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

mU eNjAN “b;”r “;”K7Fb1M.

c IN 2@bUGkPlF$k040iU (complete graph)O

?@lDN.,r}D.

}, nDN:@r}D040iUO 1

2n(n− 1)\NUr}D.

j} G, nDN@H k DN.,rbDk<WNJ$0iUNH-,

GNUN\t mO!N0r~?9.

n− k ≤ m ≤1

2(n− k)(n− k + 1)

O nDN@r}D0iUK 1

2(n− 1)(n− 2)+1\JeNU,"lP"k.

Page 11: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

3. *$i<&0iU

jA "k0iU GN9YFNUr^`D8?.;,"kH-, Gr*$i<&0iU (Eulerian graph)H$$,

=N.;r*$i<.; (Eulerian trail)H$&.

c ,*$i<.;

jA "k0iU N9YFNUr^`.;,"kH- r

>*$i<&0iU H$&

c

Page 12: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

3. *$i<&0iU

jA "k0iU GN9YFNUr^`D8?.;,"kH-, Gr*$i<&0iU (Eulerian graph)H$$,

=N.;r*$i<.; (Eulerian trail)H$&.

c 1 → 2 → 3 → 4 → 5 → 6 → 3 → 7 → 1,*$i<.;.

1

2

3

4

5

67

jA "k0iU N9YFNUr^`.;,"kH- r

>*$i<&0iU H$&

c

Page 13: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

3. *$i<&0iU

jA "k0iU GN9YFNUr^`D8?.;,"kH-, Gr*$i<&0iU (Eulerian graph)H$$,

=N.;r*$i<.; (Eulerian trail)H$&.

c 1 → 2 → 3 → 4 → 5 → 6 → 3 → 7 → 1,*$i<.;.

1

2

3

4

5

67

jA "k0iU GN9YFNUr^`.;,"kH-, Gr>*$i<&0iU (semi-Eulerian graph)H$&.

c12

3 4 5

Page 14: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

3a. *$i<&0iUN-A

}, 0iU GN4FN:@N!t, 2JeNH-, GKOD),"k.

j} / "k0iU ,*$i<&0iUG"k

,W=,roO N:@N!t,9YFvtG"k3HG"k

j} *$i<&0iUN N*$i<.;O!N=.}!GG-k

$UN@ +iP/7F !N,'K>&Bj+3KUr?Il

?IC?UO|n;h b7I)@,88?i=lb|n;h

INJ,Gb>K?IkU,J$lgJ0O6r?IkJ

?@76HO |n7?i0iU,s"kKJkUN3HG"k

Page 15: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

3a. *$i<&0iUN-A

}, 0iU GN4FN:@N!t, 2JeNH-, GKOD),"k.

j} (Euler, 1736/)"k0iU G,*$i<&0iUG"k,W=,roO GN:@N!t,9YFvtG"k3HG"k.

j} *$i<&0iUN N*$i<.;O!N=.}!GG-k

$UN@ +iP/7F !N,'K>&Bj+3KUr?Il

?IC?UO|n;h b7I)@,88?i=lb|n;h

INJ,Gb>K?IkU,J$lgJ0O6r?IkJ

?@76HO |n7?i0iU,s"kKJkUN3HG"k

Page 16: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

3a. *$i<&0iUN-A

}, 0iU GN4FN:@N!t, 2JeNH-, GKOD),"k.

j} (Euler, 1736/)"k0iU G,*$i<&0iUG"k,W=,roO GN:@N!t,9YFvtG"k3HG"k.

j} *$i<&0iUN GN*$i<.;O!N=.}!GG-k.

• $UN@ u+iP/7F,!N,'K>&Bj+3KUr?Il.

1. ?IC?UO|n;h. b7I)@,88?i=lb|n;h.

2. INJ,Gb>K?IkU,J$lgJ0O6r?IkJ.

?@76HO,|n7?i0iU,s"kKJkUN3HG"k.

Page 17: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

4. O_kHs&0iU

jA "k0iU G,,9YFN:@rlY@1LkD)r}DH-,

GrO_kHs&0iU (Hamiltonian graph)H$$,

=ND)rO_kHsD) (Hamiltonian cycle)H$&.

c 1 → 2 → 3 → 4,O_kHsD)

1

2 3

4

jA "k0iU G,,9YFN:@rlY@1Lk;r}DH-,

Gr>O_kHs&0iU (semi-Hamiltonian graph)H$&.

c

1

2 3

4

Page 18: I118 Graphs and Automatat-nemoto/teaching/2013-1-1/l5.pdf · I118 Graphs and Automata Takako Nemoto œt-nemoto/teaching/2013-1-1.html April 26

4a. O_kHs&0iUN-A

j} 0iU G, n(≥ 3)DN@rbDH9k.

GN:@ v N!t deg(v)KD$F,Y\7J$$UN 2@ v, w G

deg(v) + deg(w) ≥ n

,.)9kH-, GOO_kHs&0iUG"k.

O (Dirac, 1952/) n(≥ 3)DN@rbD0iU GN$UN 2@ v, w G

deg(v) + deg(w) ≥ n

,.)9kH-, GOO_kHs&0iUG"k.

}, st0iU G,qtDN@rbDH-, GOO_kHs&0iUGOJ$3Hr(;.

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

"k4j:`O NP04.