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2.4.5. G = (V, E)

D

_G⁰ _{= (V}⁰_{, E}⁰₎

îѲÇ

_,

J

_V⁰ _{⊆ V}

/

_E⁰ _{⊆ E,}

†

_G⁰

˚Ñ

_G

íø_²äÇ

(directed subgraph)

ì2

_2.4.6.

I

_{G = (V, E)}

uø_²Ç

_,

/

_{v0, v1}, · · · , vk ∈ V ,

†

_G

2íø‘¥

−

_(walk)

uø_Ìݲå

_{v0, e1, v1, e2, v2}, · · · , vk−1, ek, vk,

w2®á}ÑÝõ D²i5>˜

_,

U)ç

i = 1, 2, · · · , k

v

_{, ei = hvi−1, vii;}

7

_ei

˚Ñ¥−

_W

í² i

_,

w²i5,b¹Ñ¥−

_W

íÅ

_(length),

pT

_{L(W ), v0}

D

_vk

}˚Ñ¥−

W

í–õDõ

_,

1ø,H¥−pÑ

_{W = hv0, v1, v2}, · · · , vk−1, vki

Åj

_2.4.7.

ì2

_2.4.6

2

_{, W}

uâÝõ¸²iF$Aí¥−

_,

I

_{VW = {v0, v1, v2, · · ·} , vk−1, vk}

[ýwÝõFAíÕ¯

, EW = {e1, e2, · · · , ek}

[ýw²iFAíÕ¯

,

Ĥ¥−

_W

Ñø²Ç

_,

pT

_{W = (VW, EW),}

w2

_{VW ⊆ V}

7/

_{EW ⊆ E,}

FJ

_G

2©ø¥−îÑ

_G

í²äÇ

ì2

_2.4.8.

J¥−

_{W = hui,}

†

_W

˚Ñép

_(trivial)

¥−

_,

.uépí¥−˚ÑÝ ép

(nontrivial)

¥−

Ñ7Zk(/qñ5ÅH

_,

I

W(vi, vj; G)

[ýÇ$

_G

2J

_vi

Ñ–õ

_{, vj}

Ñõ íF¥−F$AíÕ¯
ép¥−uÉø_Ýõ

_u

F$Aí¥−

_,

wÅÑ

_0,

Ä

¤

W(u, u; G)

ÑÝ˛Õ¯

ì2

_2.4.9.

cq

_{W = hv0, v1, v2}, · · · , vn−1, vni

ѲÇ

_{G = (V, E)}

2í¥−

_,

J

v0 = vn,

†

_W

˚Ñ¥£í

_(closed)

¥−

ì2

_2.4.10.

J

_{W = (VW, EW)}

ѲÇ

_{G = (V, E)}

2íø‘ÝõÌóæí¥−

,

?¹

_{L(W ) = #VW − 1,}

†

_W

˚Ñ

_G

2íø‘˜

_(path),

w2¯U

_#

[ýÕ¯j Ö5_b

ì2

_2.4.11.

²Ç

_{G = (V, E)}

2

_,

J

_{W = (VW, EW)}

Ñø‘Ýépí¥£¥−

_,

/ wF²iÌóæ

_,

?¹

_{L(W ) = #EW ,}

†

_W

˚Ñø_=

_(cycle);

7³=

í²Ç$

_,

˚ÑÝ=²Ç

(directed acyclic graph)

Åj

_2.4.12.

Ý=²Ç2

_,

©ø¥−îÑø‘˜

ÄÑ

_LFG

ÑÝ=²Ç

_,

FJ…d3b‡úÝ=²ÇªWn

,

Ou(/

íqñ׶}ú²Ç7kEÍA

_,

ÝBk<ì2CìÜ?_àkݲÇ

;

Í7 Ñ7ÅH5Z

,

ÎÝvqñc_àkÝ=²ÇCݲÇ

,

´†-d2ÌJ²Ç V˚5
ÇÕâkÝ=²Ç2

_,

©ø¥−îÑø‘˜

_,

Ĥd2n˜óÉq ñvEJ¥−˚5

ì2

_2.4.13.

q

G = (V, E), G⁰ = (V⁰, E⁰)

Ñù_²Ç

_,

I

_{G = (V ∪ V}⁰_{, E ∪ E}⁰_),

†˚

_G

Ñ

_G

¸

_G⁰

í!¯

_(union),

pT

: G = G ∪ G⁰;

J

_{V ∩ V}⁰ _{6= φ,}

†I

G = (V ∩ V⁰, E ∩ E⁰),

˚

_G

Ñ

_G

¸

_G⁰

íó>

(intersection),

pT

: G = G ∩ G⁰

ì2

_2.4.14.

²Ç

_{G = (V, E)}

2

_,

cq

_{V = {v1, v2}, · · · , vn},

/

aij =





1 ,

J

_{hvi, vji ∈ E;}

0 ,

J

_{hvi, vji /∈ E;}

†

_{AG= [aij]n×n}

˚Ñ

_G

í¹Qä³

(adjacency matrix)

ì2

_2.4.15.

cq

_{G = (V, E)}

ѲÇ

_,

 Î

_G

2©_²iíj²5(

_,

JßÞ

½µíi

,

†\Gø_i

,

7tÎw…½µíi

,

|()ƒÝ²Ç

_G,

†

G

˚Ñ

_G

í

!…Ç

(underlying graph)

ì2

_2.4.16.

ø_²Çí!…Çuø_Ìj²4íÇ$

_,

˚ÑݲÇ

_,

C˚ÑÇ

$

ì2

_2.4.17.

q

G

ѲÇ

_{G = (V, E)}

í!…Ç

_,

J

_{W(vi, vj}; G) 6= φ, ∀vi, vj ∈ V ,

†

_G

˚Ñ©¦í

(connected or weakly connected)

ìÜ

2.4.18. (Chartrand & Lesniak, 1986)

I

_AG

Ñ

_{G = (V, E)}

í¹Qä³

_,

w2

_{V = {v1, v2}, · · · , vn},

/

_A^k_{G = [a}^(k)_{ij ]n×n , k ≥ 0, A}⁰_{G = In,}

†

_a^(k)_ij

[ý

W(vi, vj; G)

2ÅÑ

_k

í¥−_b

ü LFT l}Ü£wõWzp

Shavelson(1972)

D1+±6

₍₁₉₈₁₎

*¾“hõ}T|.°íÇ$l}j¶

_,

Ou‡6É5?L<ù_ÝõÈí×Vì¾Ç}

,

7I¤ù_Ýõ2È%¬5Ý õÑS

_;

(6JLù_ÝõÈiCÌiVì¾#}

,

º„5?j²4

_;

Ä7ù6úø

…!ZÇ$5l}æÊ.¯Üí°}½æ

₍

−ëyA

, 1997;

−ëy

, 2004)

n…ç 6−ëyA

(1997)

ûp

_LFG

2ù_ÝõÈF$A©!5”íÉ[ ²ií½b



(importance index)

£ù_

_LFG

钎NDĨ

(reachability measure),

ê

|hl}Ü

_,

˚T

_LFT

É

_LFT

l}5ܶ}

_,

n…ç6−ëy`¤˛‡úÉœ4ìÜ‹J„p

(

− ëyA

, 1997; Takeya, 1999),

Åqû˝6

₍

±cšA

_,

¬

_93a;

𣏠{C -rÙ&

,

¬

₉₃₎

Î7ªø¥„pw!…ìÜÕ

_,

y‡ú¤Ü54”Jbçj¶

Ì„p

,

cø,Hd.2«n5ì2£ìÜcÜà-

:

ø ì2¶}

_:

ì2

_2.5.1.

²Ç

_{G = (V, E)}

2

_,

J

_vi

Ñ–õ

_{, vj}

ÑõíF¥−F$AíÕ¯

W(vi, vj; G),

˚5Ñ©!

ì2

_2.5.2.

²Ç

_{G = (V, E)}

2

_,

J

_{vi, vj, vk, vl ∈ V ,}

/æÊ¥−

W = hvi, · · · , vk, vl, · · · , vji ∈ W(vi, vj; G),

†²i

hvk, vli

˚Ñ

_{W(vi, vj; G)}

í%âi

à‹æÊ¥−

_{W = hvi}, · · · , vk, vl, · · · , vji ∈ W(vi, vj; G),

†˚¥−

_W

%â

hvk, vli,

Cuz

_{hvk, vli}

ú

_{W(vi, vj; G)}

õ.
²Ç

G = (V, E)

2

_,

%â

_{hvk, vli}

íF©!w–õ¸õíjÖúF$A5Õ¯pT

_{C(vk, vl; G);}

?¹

C(vk, vl; G) ≡





{(vi, vj) : hvk, vli

u

_{W(vi, vj; G)}

í%âi

_{} ,}

J

_{hvk, vli ∈ E,}

φ ,

J

_{hvk, vli /∈ E,}

1J

_{C(vk, vl; G)}

íjÖ_b[ý

_{hvk, vli}

ʲÇ

_G

2íõ.
7¦

_G

2F

²i5|×õ.

_,

ªø©ø²iíõ.£d“

(normalization)

(

_,

Tà-5ì 2

ì2

_2.5.3.

²Ç

_{G = (V, E)}

2

_,

q

_{#V = n}

/

_{vk, vl ∈ V ,}

†²i

hvk, vli

5½ b

_{I(vk, vl; G)}

íì2Ñ

I(vk, vl; G) = #C(vk, vl; G)

max(vk,vl)∈V ×V #C(vk, vl; G). (2.5)

ì2

_2.5.4.

²Ç

_{G = (V, E)}

2

_,

J

_{hvk, vli ∈ E,}

/

_{#W(vk, vl; G) = 1,}

†˚

hvk, vli

Ñ!…i

(fundamental edge);

óú7k

_,

.u!…i5²i˚ÑÝ!…i

(non-fundamental edge)

ì2

_2.5.5.

²Ç

_{G = (V, E)}

2

_,

à‹

_{W(vi, vj; G) 6= φ,}

†˚

_vi

Ñ

_vj

ílWÝõ

, vj

Ñ

_vi

íªƒ®Ýõ

_;

7/

_,

I

A(vj) = {vi : W(vi, vj; G) 6= φ}, R(vi) = {vj : W(vi, vj; G) 6= φ},

†˚

_A(vj)

Ñ

_vj

5lWÝõÕ¯

, R(vi)

Ñ

_vi

5ªƒ®ÝõÕ¯

ì2

_2.5.6.

²Ç

_{G = (V, E)}

2

_,

cq

_{V = {v1, v2}, · · · , vn},

/

rij =





1 ,

J

W(vi, vj; G) 6= φ;

0 ,

J

W(vi, vj; G) = φ;

†

_{RG= [rij]n×n}

˚Ñ

_G

íªƒ®ä³

(reachability matrix)

ì2

_2.5.7.

²Ç

_{G = (V, E)}

2

_{, vi, vj, vk, vl ∈ V ,}

ì2

_{W(vi, vj; G)}

5%âiÕ¯

T (vi, vj; G)

Ñ

_:

T (vi, vj; G) ≡ {hvk, vli ∈ E : hvk, vli

Ñ

_{W(vi, vj; G)}

5%âi

},

1˚

_{#T (vi, vj; G)}

ÑÝõ

_vi

¸

_vj

Ê

_G

25©!M

(connected value)

â,Hì2ªø

,

²Ç

_G

2

_vi

¸

_vj

ù_ÝõÈ5©!MuN

:

–õÑ

_vi

/

õÑ

_vj

íF¥−w²i:Õ5ib

_,

FJ

_{T (vi, vj)}

ªeÑ

W(vi, vj; G)

í©!˙

5”í[Û

¸W

_2.5.8.

JÇ

_2.7

ÑW

_,

zp.°Ç$2óú@sÝõ5©!í%âiÕ¯

:

Gb

1

2 3 4

5

6

7 8

9

Ga Gc Gd

1

2 3 4

5

6

7 8

9

1

2 3 4

5

6

7 8

9

1

2 3 4

5

6

7 8

9

Ç

_2.7:

J

_LFT

hõªœ²Ç5tÇW

(

ùA

_:

−ëyA

, 1997)

T (v9, v1; Ga) = {hv9, v6i, hv9, v7i, hv9, v8i, hv8, v6i, hv7, v6i, hv6, v5i, hv5, v2i, hv5, v3i, hv5, v4i, hv4, v1i, hv3, v1i, hv2, v1i},

T (v9, v1; Gb) = φ,

T (v9, v1; Gc) = {hv9, v6i, hv9, v8i, hv8, v6i, hv6, v5i, hv5, v2i, hv5, v3i, hv5, v4i, hv4, v1i, hv3, v1i, hv2, v1i},

T (v9, v1; Gd) = {hv9, v7i, hv9, v8i, hv8, v6i, hv7, v6i, hv6, v5i, hv5, v2i, hv5, v3i, hv5, v4i, hv4, v1i, hv3, v1i, hv2, v1i},

FJªœ.°Ç$2óú@sÝõ5©!í%âiÕ¯ª)ƒ”íÏæ

, Takeya

(1999)

ªø¥øvÏ澓

_,

1ì2.°Ç$2óú@sÝõ5©!í%âiÕ¯5

óN

ì2

_2.5.9.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

Ñù_²Ç

_,

/

_{vi, vj ∈ V ,}

†

W(vi, vj; G)

D

W(vi, vj; G⁰)

í%âiÕ¯5óN

_{Q(vi, vj; G, G}⁰₎

ì2Ñ

Q(vi, vj; G, G⁰) = #[T (vi, vj; G) ∩ T (vi, vj; G⁰)].

ì2

_2.5.10.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

îѲÇ

_,

/

_{V = {v1, v2}, · · · , vn},

†

G

D

_G⁰

5éN

_{S(G, G}⁰₎

ì2Ñ

S(G, G⁰) = Pn

i=1

Pn

j=1#[T (vi, vj; G) ∩ T (vi, vj; G⁰)]

Pn i=1

Pn

j=1#[T (vi, vj; G) ∪ T (vi, vj; G⁰)]. (2.6)

ì2

_2.5.11.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

Ñù_²Ç

_,

†

_G

D

_G⁰

5Ĩ

R(G, G⁰)

ì2Ñ

R(G, G⁰) = 100 ×p

S(G, G⁰). (2.7)

ÇÕ

_,

−ëyA

(1997)

‡úÇ$

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

2óú@ù _Ýõ

vi



_vj

wÉ[5½b4ÊÇ$!Z,íÏæ

_,

T|¾“íj¶

_;

¹l½b

I(vi, vj; G)

D

_{I(vi, vj; G}⁰₎

íj4Ï

_,

1Y¤ì2

_G

¸

_G⁰

2óú@ù_ÝõwÉ[5 Ïæ

(discrepancy measure)

ì2

_2.5.12.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

Ñù_²Ç

_,

/

_{vi, vj ∈ V ,}

†

_G

¸

G⁰

2óú@ù_ÝõwÉ[5Ïæ

_{D(vi, vj; G, G}⁰₎

ì2Ñ

D(vi, vj; G, G⁰) = #[C(vi, vj; G) ∪ C(vi, vj; G⁰)] − #[C(vi, vj; G) ∩ C(vi, vj; G⁰)].

ù ìܶ}

_:

ìÜ

_2.5.13.

²Ç

_{G = (V, E)}

2

,

úL<

_{hvk, vli ∈ E,}

0

_{0 ≤ I(vk, vl; G) ≤ 1}

ìÜ

_2.5.14.

Ý=²Ç

_{G = (V, E)}

2

,

J

_{#V = n,}

†

(vk,vmax_l)∈V ×V#C(vk, vl; G) ≤ bn² 4 c.

Åj

_2.5.15.

擆

_2.5.14

2

_,

J

_{G = (V, E)}

uø_©¦Ç$

_,

/æÊ

_{hvk, vli ∈ E}

U )

_{#A(vk) = b}ⁿ

2c

C

_{#R(vl) = b}ⁿ

2c,

†

_{max(vk,vl)∈V ×V #C(vk, vl; G) = b}ⁿ²

4c

FJÝ=²Ç2

_,

²ií|×õ.

_,

¹

_{max(vk,vl)∈V ×V #C(vk, vl; G),}

„ .?/qíl7)

_,

ÖÍvbM3bàJ£d“®²iíõ.

_,

7DéN ƒ

®5lÌÉ

_;

OuÊ}&!‹5j„,ºxw½b4

_,

Ĥ.bql˙Vl

5
7−ëyA

₍₁₉₉₇₎

ÇJ

_{I(vk, vl; G) =} ^{#A(vk)×#R(vl)}

bⁿ²₄ c

Fì2²i5½b

_,

c_àkÅj

2.5.15

FH‘K-5Ç$

_,

ÿÜ5Ãã7k

_,

vì2.ÝÜ;

¸W

_2.5.16.

JÇ

_2.7

2²Ç

_Ga

ÑW

_,

%â²i

hv6, v5i



_{hv7, v6i}

C

_{hv9, v6i}

5©

!

_,

w–õ¸õíjÖúF$A5Õ¯}Ñ

_:

C(v6, v5; Ga) = {(v6, v5), (v6, v4), (v6, v3), (v6, v2), (v6, v1), (v7, v5), (v7, v4), (v7, v3), (v₇, v₂), (v₇, v₁), (v₈, v₅), (v₈, v₄), (v₈, v₃), (v₈, v₂), (v₈, v₁), (v₉, v₅), (v9, v4), (v9, v3), (v9, v2), (v9, v1)}.

C(v₇, v₆; Ga) = {(v₇, v₆), (v₇, v₅), (v₇, v₄), (v₇, v₃), (v₇, v₂), (v₇, v₁), (v₉, v₆), (v₉, v₅), (v9, v4), (v9, v3), (v9, v2), (v9, v1)}.

C(v₉, v₆; Ga) = {(v₉, v₆), (v₉, v₅), (v₉, v₄), (v₉, v₃), (v₉, v₂), (v₉, v₁)}.

wŸ

_,

²i

hv6, v5i



_{hv7, v6i}

C

_{hv9, v6i}

½b5ø¶à-

_:

ÄÑÇ

_2.7

2

_Ga

¯¯Åj

2.5.15

FHÇ$5‘K

_,

FJ

(vk,vmaxl)∈V ×V #C(vk, vl; G) = b9²

4c = 20, I(v6, v5; Ga) = #C(6, 5; Ga)

20 = 20

20 = 1.0, I(v7, v6; Ga) = #C(7, 6; Ga)

20 = 12

20 = 0.6, I(v9, v6; Ga) = #C(9, 6; Ga)

20 = 6

20 = 0.3.

*¸W

_2.5.16

ªø

_{I(v6, v5; Ga) > I(v7, v6; Ga) > I(v9, v6; Ga),}

¹óæ5²i

ìÜ

_2.5.18.

²Ç

_{G = (V, E)}

2

,

J

_{#V = n,}

†

Xn

i=1

Xn j=1

#C(vi, vj; G) = Xn

i=1

Xn j=1

#T (vi, vj; G). (2.9)

ìÜ

_2.5.19.

²Ç

_{G = (V, E)}

2

,

J

_{vi, vj, vk, vl ∈ V , (vk, vl) 6= (vi, vj),}

/æÊ

¥−

_{W = hvi}, · · · , vk, · · · , vl, · · · , vji,

†

_{T (vk, vl}; G) ⊂ T (vi, vj; G)

ìÜ

_2.5.20.

²Ç

_{G = (V, E)}

2

,

JæÊÝ!…i

hvi, vji

/

_{hvk, vli ∈ T (vi, vj; G),}

Ou

_{(vk, vl) 6= (vi, vj),}

†

C(vi, vj; G) ⊂ C(vk, vl; G), I(vi, vj; G) < I(vk, vl; G).

ìÜ

_2.5.21.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

Ñù_²Ç

,

J

_vi, vj ∈ V

/

_E⁰ _{⊆ E,}

†

T (vi, vj; G⁰) ⊆ T (vi, vj; G), C(vi, vj; G⁰) ⊆ C(vi, vj; G).

#T (vi, vj; G⁰) ≤ #T (vi, vj; G), #C(vi, vj; G⁰) ≤ #C(vi, vj; G).

ìÜ

_2.5.22.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

îѲÇ

,

†

₍₁₎



₍₂₎



₍₃₎

Ñg

_: (1)

úL<

_{vi, vj ∈ V , T (vi, vj}; G) = T (vi, vj; G⁰),

(2)

úL<

_{vi, vj ∈ V , C(vi, vj; G) = C(vi, vj; G}⁰_), (3) G = G⁰

ÍÜ

_2.5.23.

q

_{G = (V, E)}

D

_G⁰ _{= (V, E}⁰₎

Ñù_²Ç

,

†

₍₁₎



₍₂₎



₍₃₎

Ñ

g

_:

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[3.4:JLFT-extended°Ç3.2GaDÇ3.2Gc2óú@©!5%â¥−Õ¯ vi,vjW(vi,vj;Ga)W(vi,vj;Gc) v2,v1{hv2,v1i}{hv2,v1i} v3,v1{hv3,v2,v1i}φ v4,v1{hv4,v2,v1i}{hv4,v2,v1i} v5,v1{hv5,v2,v1i,hv5,v3,v2,v1i,hv5,v4,v2,v1i}{hv5,v2,v1i,hv5,v4,v2,v1i} v6,v1{hv6,v5,v2,v1i,hv6,v5,v3,v2,v1i,hv6,v5,v4,v2,v1i}{hv6,v5,v2,v1i,hv6,v5,v4,v2,v1i} v3,v2{hv3,v2i}φ v4,v2{hv4,v2i}{hv4,v2i} v5,v2{hv5,v2i,hv5,v3,v2i,hv5,v4,v2i}{hv5,v2i,hv5,v4,v2i} v6,v2{hv6,v5,v2i,hv6,v5,v3,v2i,hv6,v5,v4,v2i}{hv6,v5,v2i,hv6,v5,v4,v2i} v5,v3{hv5,v3i}{hv5,v3i} v6,v3{hv6,v5,v3i}{hv6,v5,v3i} v5,v4{hv5,v4i}{hv5,v4i} v6,v4{hv6,v5,v4i}{hv6,v5,v4i} v6,v5{hv6,v5i}{hv6,v5i} *wìóú@©!5%â¥−Õ¯îÑ˛Õ¯

[3.5:JLFT-extended°Ç3.2GaDÇ3.2Gc2óú@©!5%â¥−Õ¯í>Õ¸:Õ ÝõW(vi,vj;Ga)∩W(vi,vj;Gc)W(vi,vj;Ga)∪W(vi,vj;Gc) vi,vjÕ¯¥−Ÿկ¥−Ÿ v2,v1{hv2,v1i}1{hv2,v1i}1 v3,v1φ0{hv3,v2,v1i}2 v4,v1{hv4,v2,v1i}2{hv4,v2,v1i}2 v5,v1{hv5,v2,v1i,hv5,v4,v2,v1i}5{hv5,v2,v1i,hv5,v3,v2,v1i,hv5,v4,v2,v1i}8 v6,v1{hv6,v5,v2,v1i,hv6,v5,v4,v2,v1i}7{hv6,v5,v2,v1i,hv6,v5,v3,v2,v1i,hv6,v5,v4,v2,v1i}11 v3,v2φ0{hv3,v2i}1 v4,v2{hv4,v2i}1{hv4,v2i}1 v5,v2{hv5,v2i,hv5,v4,v2i}3{hv5,v2i,hv5,v3,v2i,hv5,v4,v2i}5 v6,v2{hv6,v5,v2i,hv6,v5,v4,v2i}5{hv6,v5,v2i,hv6,v5,v3,v2i,hv6,v5,v4,v2i}8 v5,v3{hv5,v3i}1{hv5,v3i}1 v6,v3{hv6,v5,v3i}2{hv6,v5,v3i}2 v5,v4{hv5,v4i}1{hv5,v4i}1 v6,v4{hv6,v5,v4i}2{hv6,v5,v4i}2 v6,v5{hv6,v5i}1{hv6,v5i}1 ¥−Å,¸P6 i=1P6 j=1P W∈W(vi,vj;Ga)∩W(vi,vj;Gc)L(W)=31,P6 i=1P6 j=1P W∈W(vi,vj;Ga)∪W(vi,vj;Gc)L(W)=46, éN:S(Ga,Gc)=31 46,ƒ®:R(Ga,Gc)=100×q 31 46=82. *wìóú@©!5%â¥−Õ¯5>ÕD:ÕîÑ˛Õ¯,¥−ŸÑ0

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[3.7:JLFT°Ç3.2GaDÇ3.2Gc2óú@©!í%âiÕ¯5>ÕD:Õ ÝõT(vi,vj;Ga)∩T(vi,vj;Gc)T(vi,vj;Ga)∪T(vi,vj;Gc) vi,vjÕ¯×üÕ¯×ü v2,v1{(v2,v1)}1{(v2,v1)}1 v3,v1φ0{(v3,v2),(v2,v1)}2 v4,v1{(v4,v2),(v2,v1)}2{(v4,v2),(v2,v1)}2 v5,v1{(v5,v4),(v5,v2),(v4,v2),(v2,v1)}4{(v5,v4),(v5,v3),(v5,v2),(v4,v2),(v3,v2),(v2,v1)}6 v6,v1{(v6,v5),(v5,v4),(v5,v2),(v4,v2),(v2,v1)}5{(v6,v5),(v5,v4),(v5,v3),(v5,v2),(v4,v2),(v3,v2),(v2,v1)}7 v3,v2φ0{(v3,v2)}1 v4,v2{(v4,v2)}1{(v4,v2)}1 v5,v2{(v5,v4),(v5,v2),(v4,v2)}3{(v5,v4),(v5,v3),(v5,v2),(v4,v2),(v3,v2)}5 v6,v2{(v6,v5),(v5,v4),(v5,v2),(v4,v2)}4{(v6,v5),(v5,v4),(v5,v3),(v5,v2),(v4,v2),(v3,v2)}6 v5,v3{(v5,v3)}1{(v5,v3)}1 v6,v3{(v6,v5),(v5,v3)}2{(v6,v5),(v5,v3)}2 v5,v4{(v5,v4)}1{(v5,v4)}1 v6,v4{(v6,v5),(v5,v4)}2{(v6,v5),(v5,v4)}2 v6,v5{(v6,v5)}1{(v6,v5)}1 Õ¯×ü,¸P6 i=1P6 j=1#[T(vi,vj;Ga)∩T(vi,vj;Gc)]=27,P6 i=1P6 j=1#[T(vi,vj;Ga)∪T(vi,vj;Gc)]=38, éN:S(Ga,Gc)=27 38,ƒ®:R(Ga,Gc)=100×q 27 38=84. *wìóú@©!í%âiÕ¯5>ÕD:ÕîÑ˛Õ¯,Õ¯×üÑ0

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Cells(x + 3, i + 4) = i: Cells(i + x + 3, 4) = i Next i

Cells(1, 4) = ”



”: Cells(x + 3, 4) = ”

Þ

_” For i = 1 To x

For j = 1 To x

Cells(i + 1, j + 4) = 0

Cells(i + 1, j + 4).Interior.Color = RGB(255, 0, 0) Cells(i + x + 3, j + 4) = 0

Cells(i + x + 3, j + 4).Interior.Color = RGB(255, 255, 0) Next j

Next i

Range(”E2”).Select End Sub

’LFT-extended

l}˙˙å5ù

_:

’epd†

Sub guide of input()

MsgBox ”

’epd†

:” & vbCrLf &

”” & vbCrLf &

”•

Hl揝p ˘`˙

_LFG

í¹Qä³

_,

ôHl揝p ˘çÞ˙

_LFG

í¹Qä

³

” & vbCrLf &

”” & vbCrLf &

”•

J²i ˘–1

_2−→

–1

₅

˙

_,

†ä³5ùüWíjÖMÑ

₁

” & vbCrLf &

”” & vbCrLf &

”•

¹Qä³íjÖMÉ?Ñ

₀

C

_1,

/.ª˛ë

” & vbCrLf &

”” & vbCrLf &

”•

p>Œ©!v

_,

ilp`DçÞu°5>Œ©!

_,

1Yó°ßåp

_” End Sub

’LFT-extended

l}˙˙å5ú

_:

p>Œ©!_b

Sub cross link()

Dim y1%, y2%, z1%, z2%

For j = 1 To x For i = 1 To x

If Cells(i + 1, j + 4) = ”” Then

y1 = y1 + 1: Cells(i + 1, j + 4).Interior.Color = RGB(255, 255, 255)

ElseIf Cells(i + 1, j + 4) <> 0 And Cells(i + 1, j + 4) <> 1 And Cells(i + 1, j + 4) <> ”” Then

y2 = y2 + 1: Cells(i + 1, j + 4).Interior.Color = RGB(0, 255, 0) Else

Cells(i + 1, j + 4).Interior.Color = RGB(255, 0, 0) End If

If Cells(i + x + 3, j + 4) = ”” Then

z1 = z1 + 1: Cells(i + x + 3, j + 4).Interior.Color = RGB(255, 255, 255) ElseIf Cells(i + x + 3, j + 4) <> 0 And Cells(i + x + 3, j + 4) <> 1 And

Cells(i + x + 3, j + 4) <> ”” Then

z2 = z2 + 1: Cells(i + x + 3, j + 4).Interior.Color = RGB(0, 255, 0) Else

Cells(i + x + 3, j + 4).Interior.Color = RGB(255, 255, 0) End If

If Cells(i + 1, j + 4) = 0 Then x1 = x1 + 1

End If Next i Next j

If x = 0 Or x1 = x * x Then

Beep: MsgBox ”

þ„p’e

_,

̶l

_,

~lp’e

”: Call clear: End End If

If y1 <> 0 Or z1 <> 0 Or y2 <> 0 Or z2 <> 0 Then Beep

MsgBox ”

˘`˙ í¹Qä³

” & y1 & ”

˛ë

,” & y2 & ”

˜Ï

;”

& vbCrLf & ”

˘çÞ˙ í¹Qä³

” & z1 & ”

˛ë

,” & z2 & ”

˜Ï

,

~½hp

_” Exit Sub

End If bb:

tmp1 = Application.InputBox(”

~p ˘`˙ 5>Œ©!í_b

_{”, ”}

p ˘`˙ 5

>Ω!_b

”, 0, Type:=1) cl = tmp1

If cl - Fix(cl) <> 0 Then

MsgBox ”

p˜Ï

,

~½hp

!”: GoTo bb Else

ReDim A(x + 2 * cl, x + 2 * cl, x + 2 * cl) For j = 1 To x

For i = 1 To x

A(i, j, 1) = Cells(i + 1, j + 4): sum1 = sum1 + A(i, j, 1) Next i

Next j

If sum1 <> 0 And cl <= sum1 Then Call cross T

Else

MsgBox ”

p˜Ï

_,

~½hp

!” & vbCrLf &

”

>Œ©!b@ükF©!b

_!”

GoTo bb End If

End If cc:

tmp2 = Application.InputBox(”

~p ˘çÞ˙ 5>Œ©!í_b

_{”, ”}

p ˘çÞ˙ 5

>Ω!_b

”, 0, Type:=1) ci = tmp2

If ci - Fix(ci) <> 0 Then

MsgBox ”

p˜Ï

_,

~½hp

!”: GoTo cc Else

ReDim B(x + 2 * ci, x + 2 * ci, x + 2 * ci) For j = 1 To x

For i = 1 To x

B(i, j, 1) = Cells(i + x + 3, j + 4): sum2 = sum2 + B(i, j, 1) Next i

Next j

If sum2 = 0 Or ci = 0 Then Exit Sub

End If

If sum2 <> 0 And ci > 0 And ci <= cl Then Call cross S

Else

MsgBox ”

p˜Ï

,

~½hp

!” & vbCrLf &

”

çÞ5>Œ©!b@ükF©!b£`í>Œ©!b

_” GoTo cc

End If End If End Sub

’LFT-extended

l}˙˙å5û

_:

p`>Œ©!1‰²

Sub cross T()

ReDim p1(cl), q1(cl) If cl = 0 Then

Exit Sub Else

For i = 1 To cl bb1:

tmp3 = Application.InputBox(”

cq ˘`˙ 5

” & i & ”

_>Œ©!Ñ ˘–1

_A”

& i & ”−→

–1

B” & i & ”

˙

_,

~p–1

A” & i & ”

íH{

:” & vbCrLf &

”

·<

_:

~ilp`DçÞu°5>Œ©!

_,

1Yåp

_{”, ”}

p ˘`˙ 5



” & i & ”

_>Ω!

”, 0, Type:=1) p1(i) = tmp3

If p1(i) < 1 Or p1(i) > x Or p1(i) - Fix(p1(i)) <> 0 Then MsgBox ”

p˜Ï

_,

~½hp

!”: GoTo bb1

End If cc1:

tmp4 = Application.InputBox(”

cq ˘`˙ 5

” & i & ”

_>Œ©!Ñ ˘–1

_A”

& i & ”−→

–1

B” & i & ”

˙

_,

~p–1

B” & i & ”

íH{

:” & vbCrLf &

”

·<

_:

~ilp`DçÞu°5>Œ©!

_,

1Yåp

_{”, ”}

p ˘`˙ 5



” & i & ”

_>Ω!

”, 0, Type:=1) q1(i) = tmp4

If q1(i) < 1 Or q1(i) > x Or q1(i) - Fix(q1(i)) <> 0 Then MsgBox ”

p˜Ï

,

~½hp

!”: GoTo cc1

End If

If A(p1(i), q1(i), 1) <> 1 Then

MsgBox ”

Fp’e.u>Œ©!

_,

~½hp

!”: GoTo bb1 End If

For k = 1 To x

If A(p1(i), k, 1) = 1 And k <> q1(i) Then A(x + 2 * i - 1, k, 1) = 1

End If

If A(k, q1(i), 1) = 1 And k <> p1(i) Then A(k, x + 2 * i, 1) = 1

End If Next k

A(x + 2 * i, x + 2 * i - 1, 1) = 1 Next i

End If End Sub

’LFT-extended

l}˙˙å5ü

_:

pçÞ>Œ©!1‰²

Sub cross S()

ReDim p2(ci), q2(ci) If ci = 0 Then

Exit Sub Else

For i = 1 To ci

bb2:

tmp5 = Application.InputBox(”

cq ˘çÞ˙ 5

” & i & ”

_>Œ©!Ñ ˘–1

_A”

& i & ”−→

–1

B” & i & ”

˙

_,

~p–1

A” & i & ”

íH{

:” & vbCrLf &

”

·<

_:

~Yp`DçÞu°5>Œ©!5ßåp

”, ”

p ˘çÞ˙ 5

_”

& i & ”

_>Ω!

”, 0, Type:=1) p2(i) = tmp5

If p2(i) < 1 Or p2(i) > x Or p2(i) - Fix(p2(i)) <> 0 Then MsgBox ”

p˜Ï

,

~½hp

!”: GoTo bb2 End If cc2:

tmp6 = Application.InputBox(”

cq ˘çÞ˙ 5

” & i & ”

_>Œ©!Ñ ˘–1

_A”

& i & ”−→

–1

B” & i & ”

˙

_,

~p–1

B” & i & ”

íH{

:” & vbCrLf &

”

·<

_:

~Yp`DçÞu°5>Œ©!5ßåp

”, ”

p ˘çÞ˙ 5

_”

& i & ”

_>Ω!

”, 0, Type:=1) q2(i) = tmp6

If q2(i) < 1 Or q2(i) > x Or q2(i) - Fix(q2(i)) <> 0 Then MsgBox ”

p˜Ï

_,

~½hp

!”: GoTo cc2

End If

If B(p2(i), q2(i), 1) <> 1 Then

MsgBox ”

Fp’e.u>Œ©!

_,

~½hp

!”: GoTo bb2 End If

For k = 1 To x

If B(p2(i), k, 1) = 1 And k <> q2(i) Then B(x + 2 * i - 1, k, 1) = 1

End If

If B(k, q2(i), 1) = 1 And k <> p2(i) Then B(k, x + 2 * i, 1) = 1

End If Next k

B(x + 2 * i, x + 2 * i - 1, 1) = 1 Next i

End If

End Sub

’LFT-extended

l}˙˙å5ý

_:

l½b

Sub importance index() Call check

For i = 1 To x + 2 For j = 1 To x + 2

Cells(i, j + x + 4).Interior.Color= RGB(192, 192, 192): Cells(i, j + x + 4) =””

Next j Next i max1 = 0

ReDim D(x, x), E(x, x) For n = 2 To x

For i = 1 To x + 2 * cl For j = 1 To x + 2 * cl

A(i, j, n) = 0

For k = 1 To x + 2 * cl

A(i, j, n) = A(i, j, n) + A(i, k, n - 1) * A(k, j, 1) Next k

Next j Next i Next n

For i = 1 To x For j = 1 To x For n = 1 To x - 1

D(i, j) = D(i, j) + A(i, j, n) If cl > 0 Then

For k = 1 To cl

If i = q1(k) And j <> p1(k) Then

D(i, j) = D(i, j) + A(x + 2 * k, j, n) End If

If j = p1(k) And i <> q1(k) Then

D(i, j) = D(i, j) + A(i, x + 2 * k - 1, n) End If

Next k End If Next n Next j Next i

For i = 1 To x For j = 1 To x

If A(i, j, x) <> 0 Or D(i, i) <> 0 Then

MsgBox ”

p˜Ï

,

ßÞ=

_,

~½hp

!”: Exit Sub End If

Next j Next i

For i = 1 To x max2 = 0

For s = 1 To x - 1 For r = 1 To x

If A(r, i, s) ¡¿ 0 Then max2 = s

End If Next r Next s

For j = 1 To x

sum1 = 0: sum2 = 0 For n = 1 To x

sum1 = sum1 + D(n, i): sum2 = sum2 + D(j, n) Next n

E(i, j) = (sum1 + 1) * A(i, j, 1) * (sum2 + 1)+ A(i, j, 1) * max2 If E(i, j) > max1 Then

max1 = E(i, j) End If

Next j Next i

For i = 1 To x

Cells(1, i + x + 6) = i: Cells(i + 1, x + 6) = i Next i

Cells(1, x + 5) = ”

½b

_” For i = 1 To x

For j = 1 To x

Cells(i + 1, j + x + 6).Interior.Color = RGB(255, 0, 255) If 100 * E(i, j) / max1 - Fix(100 * E(i, j) / max1) >= 0.5 Then

Cells(i + 1, j + x + 6) = (Fix(100 * E(i, j) / max1) + 1) / 100 Else

Cells(i + 1, j + x + 6) = Fix(100 * E(i, j) / max1) / 100 End If

Next j Next i End Sub

’LFT-extended

l}˙˙å5þ

_:

lĨ

Sub reachability measure() Dim SumA%, SumB%, SumC%

ReDim C(x + 2 * ci, x + 2 * ci, x + 2 * ci) Dim r

Call check

Range(”A22:C22”).Interior.Color = RGB(192, 192, 192) Cells(22, 1) = ””: Cells(22, 3) = ””

For j = 1 To x + 2 * ci For i = 1 To x + 2 * ci

C(i, j, 1) = A(i, j, 1) * B(i, j, 1)

Next i Next j

For n = 2 To x

For i = 1 To x + 2 * cl For j = 1 To x + 2 * cl

A(i, j, n) = 0

For k = 1 To x + 2 * cl

A(i, j, n) = A(i, j, n) + A(i, k, n - 1) * A(k, j, 1) Next k

Next j Next i Next n

For n = 2 To x

For i = 1 To x + 2 * ci For j = 1 To x + 2 * ci

B(i, j, n) = 0: C(i, j, n) = 0 For k = 1 To x + 2 * ci

B(i, j, n) = B(i, j, n) + B(i, k, n - 1) * B(k, j, 1) C(i, j, n) = C(i, j, n) + C(i, k, n - 1) * C(k, j, 1) Next k

Next j Next i Next n

For n = 1 To x - 1 For i = 1 To x + 2 * cl For j = 1 To x + 2 * cl

If (i <= x And j <= x) Or (i > x And (i - x) / 2 = (i - x) \ 2) Or (j > x And (j - x) / 2 <> (j - x) \ 2) Then

SumA = SumA + A(i, j, n) * n End If

Next j

Next i Next n

For n = 1 To x - 1 For i = 1 To x + 2 * ci For j = 1 To x + 2 * ci

If (i <= x And j <= x) Or (i > x And (i - x) / 2 = (i - x) \ 2) Or (j > x And (j - x) / 2 <> (j - x) \ 2) Then

SumB = SumB + B(i, j, n) * n: SumC = SumC + C(i, j, n) * n End If

Next j Next i Next n

For i = 1 To x max3 = 0

For s = 1 To x - 1 For r = 1 To x

If A(r, i, s) ¡¿ 0 Then max3 = s

End If Next r Next s

For j = 1 To x

SumA = SumA + A(i, j, 1) * max3 SumO = SumO + C(i, j, 1) * max3 Next j

Next i

SumA = SumA - cl: SumB = SumB - ci: SumC = SumC - ci For i = 1 To x

For j = 1 To x For n = 1 To x

If A(i, j, x) <> 0 Or A(i, i, n) <> 0 Or B(i, j, x) <> 0 Or B(i, i, n) <> 0 Then

MsgBox ”

p˜Ï

,

ßÞ=

_,

~½hp

!”: Exit Sub End If

Next n Next j Next i

Range(”A22:C22”).Interior.Color = RGB(255, 0, 255) Cells(22, 1) = ”

Ĩ

_=”

Cells(22, 3) = ””

r = 100 * Sqr((SumC + SumO) / (SumA + SumB - SumC)) If 10 * r - Fix(10 * r) ¿= 0.5 Then

Cells(22, 3) = (Fix(10 * r) + 1) / 10 Else

Cells(22, 3) = Fix(10 * r) / 10 End If

End Sub

’LFT-extended

l}˙˙å5ÿ

_:

lÝu5²iÝõÉ[5Ïæ

Sub discrepancy() Call check

For i = 1 To x + 2 For j = 1 To x + 2

Cells(i + x + 2, j + x + 4).Interior.Color = RGB(192, 192, 192) Cells(i + x + 2, j + x + 4) = ””

Next j Next i

ReDim D(x, x), E(x, x), F(x, x), G(x, x) For n = 2 To x

For i = 1 To x + 2 * cl For j = 1 To x + 2 * cl

A(i, j, n) = 0

For k = 1 To x + 2 * cl

A(i, j, n) = A(i, j, n) + A(i, k, n - 1) * A(k, j, 1) Next k

Next j Next i Next n

For n = 2 To x

For i = 1 To x + 2 * ci For j = 1 To x + 2 * ci

B(i, j, n) = 0

For k = 1 To x + 2 * ci

B(i, j, n) = B(i, j, n) + B(i, k, n - 1) * B(k, j, 1) Next k

Next j Next i Next n

For i = 1 To x For j = 1 To x For n = 1 To x - 1

D(i, j) = D(i, j) + A(i, j, n) F(i, j) = F(i, j) + B(i, j, n) If cl > 0 Then

For k = 1 To cl

If i = q1(k) And j <> p1(k) Then

D(i, j) = D(i, j) + A(x + 2 * k, j, n) End If

If j = p1(k) And i <> q1(k) Then

D(i, j) = D(i, j) + A(i, x + 2 * k - 1, n) End If

Next k End If

If ci > 0 Then

For m = 1 To ci

If i = q2(m) And j <> p2(m) Then F(i, j) = F(i, j) + B(x + 2 * m, j, n) End If

If j = p2(m) And i <> q2(m) Then

F(i, j) = F(i, j) + B(i, x + 2 * m - 1, n) End If

Next m End If Next n Next j Next i

For i = 1 To x For j = 1 To x

If A(i, j, x) <> 0 Or D(i, i) <> 0 Then

MsgBox ”

`’ep˜Ï

_,

ßÞ=

_,

~½hp

!”: Exit Sub End If

If B(i, j, x) <> 0 Or F(i, i) <> 0 Then

MsgBox ”

çÞ’ep˜Ï

_,

ßÞ=

_,

~½hp

!”: Exit Sub End If

Next j Next i

For i = 1 To x max4 = 0

For s = 1 To x - 1 For r = 1 To x

If A(r, I, s) ¡¿ 0 Then max4 = s

End If Next r Next s

For j = 1 To x

sum1 = 0: sum2 = 0: sum3 = 0: sum4 = 0 For n = 1 To x

sum1 = sum1 + D(n, i): sum2 = sum2 + D(j, n) sum3 = sum3 + F(n, i): sum4 = sum4 + F(j, n) Next n

E(i, j) = (sum1 + 1) * A(i, j, 1) * (sum2 + 1)+ A(i, j, 1) * max4 G(i, j) = (sum3 + 1) * B(i, j, 1) * (sum4 + 1)

Next j Next i

For i = 1 To x

Cells(x + 3, i + x + 6) = i Cells(i + x + 3, x + 6) = i Next i

Cells(x + 3, x + 5) = ”

Ïæ

_” For i = 1 To x

For j = 1 To x

Cells(i + x + 3, j + x + 6).Interior.Color = RGB(0, 0, 200) If A(i, j, 1) + B(i, j, 1) = 1 Then

Cells(i + x + 3, j + x + 6) = Abs(E(i, j) - G(i, j)) Else

Cells(i + x + 3, j + x + 6) = 0 End If

Next j Next i End Sub

’LFT-extended

l}˙˙å5

_:

Œu´˛p’e

Sub check() For j = 1 To x For i = 1 To x

If Cells(i + 1, j + 4) = 0 Then x1 = x1 + 1

End If Next i Next j

If x = 0 Or x1 = x * x Then

Beep: MsgBox ”

þ„p’e

_,

̶l

_,

~lp’e

”: Call clear: End End If

End Sub

’LFT-extended

l}˙˙å5

_:

ÀÎ T[£‰b¦É

Sub clear()

ReDim A(x + 2 * cl, x + 2 * cl, x + 2 * cl), B(x + 2 * ci, x + 2 * ci, x + 2 * ci), C(x + 2 * ci, x + 2 * ci, x + 2 * ci)

Range(”A1:CZ80”).Value = ””

Range(”A1:CZ80”).Interior.Color = RGB(192, 192, 192) For n = 1 To x

For i = 1 To x + 2 * cl For j = 1 To x + 2 * cl

A(i, j, n) = 0 Next j

Next i Next n

For n = 1 To x

For i = 1 To x + 2 * ci For j = 1 To x + 2 * ci

B(i, j, n) = 0: C(i, j, n) = 0 Next j

Next i Next n

x = 0: x1 = 0

End Sub

Ë“ù: LFT-extended l}ZªN™˙5ÏWåÞ

LFT-extended

l}ZªN™˙5ÏWåÞ