! #" $&%'$)(+*-,.0/1*32547698;:=<
>?6A@B8DC
>E,FHGIJG<K,L*M@NFBO1IP<RQSIB*7THUAV12WX>QY4[Z\4]QX,\G=^5_MZ`THI
4-2NabZ:Ob,Tc8<d,\Ce8gfE*A,h.iIjC
k'l[mon'pqsr
∗
t
436]O1QSI7_uIvG25>1W&,\G=O1QSIw_uIBGIjOx476y,O1Gz,\*MG2B<d8
n
{
<RQ|IB*3THUAV12 WX>}2B<d8xfE*A,h.~/I\WXO?8;: >?6-@
{
*3Ijf}2`<Q|IB*3THUAV12 WX>E,CQ4[Zs|QSUjTBa?8
1, 2, . . . , n
25TBOb,\UjTBC 8fE2438CLa2E|IjC
K
n
r
∗
,\FHG8
n
{
<RQ|IB*3THUAV12 WX>}2B<d8fE*-,./I\WXO?8Q|TH2NC25*7'UjTBO?8`_uIj436KT
K
n
r
#" $&%
! #" $" $(+*-,.
H
⊆ G
_uIj436+\E`<
fE*-,'I
G
<6-IBG8Q68D|>E2s<6AIBG8;:fEG8_uIj436
fE*A,h.iIjC
/I\WXO?8Cr
∗ ω(G) =
Cx,H
{|V (H)| : H
_uIj476>SQ>EZe<
G
}
6-2|QSUjTBab,5NwfE*A,h.#^
G
r
∗
,L>5,\FHGIjfE2fE*A,h.#^
ω
(G) ≥ 1
Q
ω
(G) ≤ ∆(G) + 1
r
∗
Q|UjTBab,L>|Q|>E2H<d,+_7Ij476/b,\*A,CIy6-*3IjC
6-*3^1G=O?8C
2Na1SQ|UjTHIjO1Q|2H<2=r
#" $"'
! #" $S¡$K(+*-,.w/1*M2N4368;:<
>;6-@5*u8C
F,G=O1IsG<d,<Q|IB*3THUAV12 WX>Q¢O1Q|Is4£Z4-Q,G=^5_MZxTHI
4-2NabZ:Ob,Tc8<d,\Ce8gfE*A,h.iIjC
k¤?¥j¦&pqsr
∗
t
436]O1QSI7_uIvG25>1W&,\G=O1QSIw_7IBGIjO§476y,O1Gz,*3G2H<8
n
{
<RQSIB*7THUAV12WX>E2H<8f}*-,.¨/^43698;:>?6-@
{
*3Ijf}2`<Q|IB*3THUAV12 WX>E,CQ4[Zs|QSUjTBa?8
1, 2, . . . , n
25TBOb,\UjTBC 8fE2438CLa2E|IjC
N
n
r
∗
,\FHG8
n
{
<RQ|IB*3THUAV12 WX>}2B<d8©fE*-,./I\WXO?8+Q|TH2NC25*7'UjTBO?8_uIj436YT
N
n
TBa1Q|@E*>*-,c<ªBG=THQ
G2H<2EO1IjfE2f}*-,h.i^/^436-IjfE2J_uIj436KTBa1QS25*3IjC
/^4368Cr
∗ K
1
∼
= N
1
2E*-,T
N
n
⊆ K
n
GX,>E,FHGIjf}2
n
r
«D¬¬S¬
! #" $~$ba1QS@5*
U
⊆ V
_uIj436PTBa1Q|2E*3IjC
n'
l
5lcn1pq
<
f}*-,'I
G
= (V, E, ψ)
<6AIBG8gQ68D|>E2<6AIBG8;:=f}G=^
U
= ∅
|^a
G
[U ]
_uIj436Kf}*-,h.&IjC
/^43698Cr
∗ α(G) =
Cx,H
{|U | : U
_7Ij476TBa1Q|2E*3IjC
O1Q|IjT,\SIjFBO?8C
<
G
}
6A2§|QSUjTBab,n1
l
lcn j<RQSIB*7THUAV12WX>E2H<I7_dfE*A,h.#^
G
r
∗ 1 ≤ α(G) ≤ |V | − δ(G)
25*-,\T
α
(G) + ω(G) ≤ |V | + 1
r
∗
Q|UjTBab,LO1Q|IjT,|IjFBO12 -UBQ_uIj436K/b,\*A,CIy6-*MIjC
6A*7^1G=O?8DC
25a1|Q|UjTHIjO1QS2B<2=r
#" $S¡b
! #" $ $(+*-,.
G
= (V, E, ψ)
_uIj436df}*-,h.&IjC
r
Al5¤D|yn1pq
AlE¤|£n1pq
¥j¦
Ak
n1o
r
<d6-IBG8
QK698D|>E2
<6-IBG8;:RfEG8
deg
(v) = r
GX,
>E,FHGIjf}2
<RQSIB*7THUAV12WX>5,
v
∈ V
r
∗
,\FHG8fE*A,h./^43698`_uIj436Kf}*-,h.&IjC
0
{
*3Ijf5^1X,\*3O?8Cr
∗
,\FHG8
n
{
<RQ|IB*3THUAV12 WX>}2B<d8fE*-,./I\WXO?8L_7Ij476RfE*-,.iIjC
(n − 1)
{
*3IjfE^1,\*3O?8Cr
∗
(J*-,h.
8
3
{
*MIjf5^1X,\*7O1I
6-2
f}*-,h.
8
\¤jcn l
Q|U[V
/2 G=fE*A,h.X8
&2
<Q|IB*7THU[V12 WX>E,\UAV
436-2N/O1QX,
0, 1, 2
^a
3
6-2fE*A,h.X8xk\\¤ccn lcr
#" $
! #" $ $"!;6y,O1Gz,*3G2H<8
fE*-,.+/1*M2N4368;:
>?6-@E*3Ijf}2
<Q|IB*3THUAV12 WX>E,CQK4[Z
<K4]Tc8436]>QSI
k
{
IBSIjCIjO;6-2H<I UBQZEf}Q¨a1QOb,\*3O1IeQ <
>?6-@E*
8C
>*A,y<ªBG=THQ|IW&Z\UjTZL698D>}26-I©4M/2 -*3@G
UBQZ}fE@H<e:>;6-@5*MIx*M@NFBO1QZ
4-Q|ª§G25>1W&,\G=O1QSI _uIBG=O?8DC
IB|IjCIjO?6AIjC:25TBOb,\UjT,C 8
Q
k
Q
Ob,Tc8<d,\Ce8
k
5¥j¦#5#9¥y¤
k'l$AEh¥j¦#E
r
∗
!^/IB*7>}2N436]>E,
Q
k
C§,
2
k
<Q|IB*3THUAV12 WX>}@B<
Q
k
2
k
−1
>*A,y<ªBG=THQor
∗
!^/IB*7>}2N436]>E,
Q
k
_uIj476fE*A,h.iIjC
k
{
*MIjf5^1X,\*7O;8Cr
∗ Q
1
∼
= K
2
r
¬&%
! #" $ $x(J*-,h.
G
= (V, E, ψ)
_7Ij476
f}*-,.iIjC
N
¤
c
lcn1pq
<6-IBG8
QJ68D|>E2
<6AIBG8;:©fEG8
Q|476-O1Q|I7_MZ
*325TW&Z\UjTBO1I
TBa1QS25*
8
O1Q|IjT,\SIjFBO1I
V
1
, V
2
⊆ V
6£,>Q|IE:©FHI
V
1
∪ V
2
= V
r
∗
2 G=THQ|Ob,
{V
1
, V
2
}
6-2
k
co\m
N
¤
c
lyn1psfE*A,h.#^
G
:D,JTBa1QS25*
8
V
1
, V
2
6-2
k
¦&p
hl
N
¤
c
lyn jfE*-,.#^
G
r
∗
2G=fE*-,.1fE*A,h.#^sG<R^1G=THQ|IBO1Ijf}2_uIj436fE*A,h.iIjC
G<K^1G=THQSIBO?8C
fE*A,h.X8LG<K^1G=THQ|IB|O1I
O1Q|IJC2Nf ZL/254]QX,\Gz,©/ªy6A|Q
r
#" $'
!/2 -*3@ G2NC@H<RQ|25O?8DUAV
G2 6AI7_/2E*
8§fE*A,h.i@H<e:=fE*-,.
,CQ G<R^1G=THQ|IBO;8CQ
4£ZCr
QOYr<K4]Tc8436]>QSI
4]^/IB*3>E25476->Qo:'f}*-,h.
8/^436-I
Q
1
{
*3Ijf5^1X,\*3O1IEr
! #" $ $!(+*A,h.~/1*32547698
G
= (V, E, ψ)
_7Ij476wk'l[mon'pq
] Hlyq
N
¤
c
lyn1pq
<6
{
IBG8Q68=>E2
<6AIBG8;:?f}G8sQ|476-O1Q|I7_uI6y,>Q'/2G=THQ,}WG<K^1G=THQSIBO?8
{V
1
, V
2
}
:EFHI>E,\FHG8
<RQSIB*7THUAV12W
Ij>
u
∈ V
1
4£Z4-Q,G=^5_uI+<
G
T
>5,\FHG8C
<Q|IB*3THUAV12 WX>QSIjC
v
∈ V
2
r
∗
t
436]O1Q|I7_7I
G2N>1W&,G=O1Q|I_uIBGIjO
436£,\O1Gz,\*MG2B<d8J/I\WXO?8Pf}*-,.=G<R^1G=THQ|IBO;8;:G,
>?6-@E*3IjfE2
V
1
= {1, 2, . . . , r}
Q
V
2
= {r +1, . . . , r +s}
2NTBOb,UjT,C 8f}2+478CLa25SIjC
K
r,s
r
∗
,FHG8
/I\WXO;8
f}*-,h.G<R^1G=THQ|IBO;8;:¨G,§>?6A@5*MIjfE2
|V
1
| = r
2E*-,\T
|V
2
| = s
_uIj436
QTH25C2E*u'UjTBO?8gT
K
r,s
r
∗ K
1
,1
∼
= K
2
25*A,T
K
2
,2
∼
= Q
2
r
#" $ b
! #" $ $
(J*-,h.
G
= (V, E, ψ)
_7Ij476
f}*-,h.&IjC
k
c
lyn1pq
<6AIBG8
Q698D|>E2
<6AIBG8;:Kf}G8
Q436]O1Q|I7_]Z
*M2NTW&Z\UjTBO1I
TBa1QS25*
8
O1Q|IjT,|IjFBO1I
V
1
, . . . , V
k
⊆ V
6£,>Q|IE:
FHI
V
1
∪ . . . ∪ V
k
= V
r
∗
(+*A,h.X8
1
{
G=THQ|IB|O1IP6-2fE*-,.X8
/^476AIEr
∗
2 G=f}*-,.
f}*-,.#^
k
{
G=THQ|IB|O1IjfE2s_uIj436©fE*-,.iIjC
k
{
G=THQ|IB|O?8C
fE*A,h.X8
k
{
G=THQ|IB|O1IsO1Q|I
C2Nf;ZL/2N4-Q,\Gz, ©/ªy6-SQor
∗
,\FHG8
n
{
<RQ|IB*3THUAV12 WX>}2B<d8©fE*-,.aIjT)/ªy6-|Qj_uIj436¨f}*-,.iIjC
k
{
G=THQ|IB|O?8C
G,/Iy<RO1IjfE2
k
≤ n
>5,\FHG8fE*A,h.
k
{
G=THQ|IB|O?8`_7Ij476fE*A,h.iIjC
l
{
G=THQ|IB|O?8C
G,`>5,\FHGIjfE2
l
≥ k
r
%
! #" $&%$(+*-,.s/1*32547698
G
= (V, E, ψ)
_uIj436
k'l[mon1pq
] Hl£q
k
c
lc
n'pq
<6AIBG8
Q68=>E2g<6-IBG8;:~fEG8
Q|476-O1Q|I7_MZgO1Q|Ij/^436-IQ
*32NTW&ZUjTBO1ITBa1Q|25*u8
O1Q|IjT,|IjFBO1I
V
1
, . . . , V
k
⊆ V
6£,>Q|IE:FHI
V
1
∪. . .∪V
k
= V
QEGX,
>E,FHGIjf}2
i
= 1, . . . , k
:
>E,FHG8
<RQSIB*7THUAV12W
Ij>
u
∈ V
i
4£Z4-Q,G=^5_uI+<
G
T
>E,\FHG8C
<Q|IB*7THU[V12 WX>QSIjC
v
∈ V \ V
i
r
∗
t
476-O1Q|I7_uIJG25>1W&,\G=O1QSI_uIBGIjO436£,\O1Gz,\*MG2B<d8g/I\WXO?8xf}*-,h.
k
{
G=THQ|IB|O?8;:DGX,>?6A@5*3Ijf}2
V
i
= {s
i
+ 1, s
i
+ 2, . . . , s
i+1
}
: f}G=THQ|I
s
1
= 0
Q
s
i+1
= s
i
+ r
i
25TBOb,\UjT,\Ce8fE2
438CLa2E|IjC
K
r
1
,r
2
,...,r
k
r
∗
,FHG8
/I\WXO?8
fE*A,h.
k
{
G=THQ|IBO?8;:GX,
>?6-@E*3Ijf}2
|V
i
| = r
i
:
i
= 1, . . . , k
: _7Ij476
QTH25C2E*u'UjTBO?8gT
K
r
1
,...,r
k
r
! #" $&%'%'$+(+*A,h.
G
= (V, E, ψ)
_uIj436n1
lA¥k
£n1p
<d6-IBG8
Q¢698D|>E2
<6AIBG8;:)f}G8
Q436]O1QSI7_MZ
O1Q|Ij/^436-IQ'*M2NTW&ZUjTBO1IR/2G=TBa1Q|25*u8
V
1
, V
2
⊆ V
6£,\>Q|IE:FHI
V
= V
1
∪ V
2
Q
>E,FHGz,LT
>*-,c<ªBG=THQYf}*-,h.i^
G
W&Z\UjTc8x<Q|IB*3THUAV12 WX>QYOb,|IjFZ\UBIe*3@H<KO12 UjTHI
]O1Q|I©G2
V
1
^a*M@B<RO12 UjTHI
]O1Q|IeG2
V
2
r
#" $'
!/2
]*M@ G
2NCx,y<Q,\O?8DUAV
G2g6-I7_+/25*u8
fE*A,h.i@H<e:f}*-,.
,CQ¢4]/@\_uO?8CQ
4[Z
Cr
QOYr)f}*-,.X8/I\WXO1IE:D4]^/IB*7>}2N436]>Q¨Q /I\WXO1IJf}*-,h.
8
k
{
G=THQSIBO1IE:
k
≥ 2
r
#" $ b
! #" $&%" $D(+*A,h./1*M2N4368
G
= (V, E, ψ)
aªBGzZ\Uy8+*M@B<RO12 UjTHI
MO1QSI
fE*A,h.iIjC
4]/@\_uO?8C
Q
2
{
*3Ijf5^1X,\*3O?8C
6-2
cpr
∗
I
-|Q
n
≥ 3
:H6A2KQ|476-O1Q|I7_uIG2N>1W&,G=O1Q|I¨_uIBGIjO
436£,O1Gz,*3G2H<8 Uy8>o:>;6-@5*MIjfE2d<Q|IB*7T
{
UAV12WX>5,\CQ
4[Z
|QSUjTBa?8
1, 2, . . . , n
:
,
>*-,c<ªBG=THQX,CQ
TBa1Q|25*u8
{1, 2}, . . . ,
{n − 1, n}, {n, 1}
2NTBOb,UjT,C 8f}2438CLa25|IjC
C
n
r
∗
,\FHG8
n
{
<Q|IB*7THU[V12 WX>E2H<8Uy8D>QTH25C2E*u'UjTBO?8v_uIj436
T
C
n
>E,FHG8sUy8>D/2N4-Q,\Gz,
/1*7Tc8DOb,
_uCO1QSI7_
3
<Q|IB*7THU[V12 WX>Q
r
∗
8>¢/2N4-Q,Gz,
_]Z\Uy8
/b,*7Tc8436£Z
|Q|UjTBaª<RQSIB*7THUAV12WX>E@H<
6A2Uy8>0kcph¥j¦&p
Uy8>
/2N4-Q,Gz,
_]Z\Uy8
O1Q|Ij/b,\*3Tc8436£Z|QSUjTBaªP<Q|IB*3THUAV12 WX>}@B<
6A2Uy8>n1
lkcph¥j¦&pNr
∗ C
3
∼
= K
3
Q
C
4
∼
= K
2
,2
r
%=¬
#" $ b
! #" $&% ¡$+(+*A,h.R/1*32N43698
aIjT§Uy8>|Q
,Uy8>|QSUjTBO?8
6A2
¥
X,4`aªBGzZ\Uy8
fE*A,h.iIjC
4]/@_
O;8C
6A2#
l£w5r
*7THIy<2436-25/O1Q,
0, 1
^a
2
6-2
-UBQ|IjFB>E,r
∗
t
476-O1Q|I7_uILG25>1W&,\G=O1QSI
_uIBG=Ob,436£,O1Gz,*3G2H<d,
-UBQ|IjFB>E,:>?6A@5*3I7_K<RQ|IB*3THUAV12 WX>E,\CQ04[Z
|QSUjTBa?8
1, 2, . . . , n
:?,>*-,c<ªBG=THQX,CQ=TBa1Q|2E*
8
{1, 2}, . . . , {n−1, n}
2NTBOb,\UjT,\C 8
fE2438CLa2E|IjC
P
n
r
∗
,\FHGz,
n
{
<RQSIB*7THUAV12WX>E2H<d,
-UBQ|IjFB>E,sQTH2NC25*7'UjTBOb,_uIj436T
P
n
r
∗ P
1
∼
= K
1
:
P
2
∼
= K
2
Q
P
3
∼
= K
2
,1
r
#" $&%
! #" $&%
~$
!/@_
O;8
/2 G=fE*A,h.f}*-,.#^
G
_uIj436_uIjf}2¥[hm
?Nwx4]/@\_uO12 -UBQ~<d6-IBG8
Q
698D>}2<6AIBG8;:'fEG8>E,FHG8_uIjfE2Ob,G=fE*-,.<+W&, -UBQ
<d8aªBGzZ\Uy8
/2G=fE*-,.iIjC
G
_7Ij476
fE*A,h.iIjC
O1QSIj4M/@_
O?8Cr
#" $&%'%D
%=¬¬
! #" $&%
$dQSIBUAV
G
aªBG=THQ|I
f}*-,h.&IjCr
b¤q
f}*-,.i@B<
G
1
, G
2
⊆ G
Ob,\Tc8
{
<K,C 86-IjO§4]/2 -*3@ G/2 G=fE*A,h.i@H<
fE*A,h.#^
G
: >?6-@IjfE2LTBa1Q|2E*3IjC
<Q|IB*3THUAV12 WX>}@B<
_7Ij476
V
(G
1
) ∪ V (G
2
)
Q>?6-@E*3Ijf}2TBa1QS25*3IjC
>*A,y<ªBG=THQ=_uIj476
E
(G
1
) ∪ E(G
2
)
r
∗
!^CªLfE*A,h.i@H<
G
1
Q
G
2
25TBOb,\UjT,\Ce8
478DCa2E|IjC
G
1
∪ G
2
_uI7_K<K,\*u6-2
+_uIj476
O1Q|IjT,|IjFBOb,s2 G<8Da2E*7^f}*-,.#^
G
r
∗
W&,\4MO12
]UBQJ4]^C 8
f}*-,h.&@B<
4£Z
/2 G2NaO1I
G2
<JW&,4]O12 -UBQ
4M^C 8
TBa1Q|2E*3@H<
<
4]THUjTHIjfE@5|O12 -UBQ_uIj4762NOb,W&Z\UjTBOb,`Q/1*3THIjCQSIjOOb,r
∗
(+*A,h.z_uIj436K4]/@\_uO?8§<6AIBG8Qb698D>}2`<d6-IBG8;:=fEG8O1QSIPC2NFBOb,fE2s/1*3THIBG=476y,y<Q
+<
/2N436£,UBQ4]^C 8fE*A,h.i@H<
2*M2NTW&Z\UjTBO;8DUAVTBa1QS25*-,UAVg<Q|IB*7THU[V12 WX>E@H<er
#" $&%"z
! #" $&%
$Y(+*A,h./1*M2N4368
G
= (V, E, ψ)
_7Ij476KE\m
lyq
<d6-IBG8Q68=>E2©<6AIBG8;:?f}G8
/2N4-Q,Gz,d<Q|IB*7THU[V12 W
Ij>
v
6£,\>Qo:}FHI
V
\{v} = N (v)
25*A,Tf}*-,.
G
−v
_uIj436¢Uy8>|IjC
∗
IjFHIBSQ
n
≥ 4
:;6-2LQ436]O1QSI7_uIJG2N>1W&,G=O1Q|I
_uIBG=O12476y,O1Gz,*3G2H<IJ>}2 W
2=:=G,e>?6A@5*3Ijf}2
V
= {1, 2, . . . , n}
:
v
= n
Q
G
− v = C
n
−1
25TBOb,\UjT,C 8L_7IJ478CLa25SIjC
W
n
r
∗
,\FHGI
n
{
<Q|IB*7THU[V12 WX>E2H<Ie>}2 W
2QTH25C2E*u'UjTBO1I_uIj436RT
W
n
r
∗ W
4
∼
= K
4
r
#" $&% ¡
%D¬¬S¬
! #" $&%
$(J*-,h.&IjC
]N
coN
pq
O1QSIj/^476AIjfE2fE*-,.#^
G
= (V, E, ψ)
Ob,\Tc8
{
<K,C 8§476y,O1Gz,\*MG2B<d8fE*A,h.!/1*M2N4368
L
(G)
:>?6-@E*3IjfE2sTBa1Q|25*MIjC
<RQSIB*7THUAV12WX>E@H<
_uIj436
E
:,G<d,s<Q|IB*3THUAV12 WX>Q
e
1
Q
e
2
4[Z\4]QX,\G=^5_MZTHIe4-2NabZs<d6-IBG8QY698D>}2<d6-IBG8;:bf}G8
>*A,y<ªBG=THQ|I
e
1
, e
2
4£Z4£Z4-Q|IBG=O1QSI+<
f}*-,'I
G
r
#" $&%
! #" $&%
$AkzlAm
n1
lyn1
l£q
fE*A,h.#^
/1*M2N436-IjfE2
G
= (V, E, ψ)
Ob,Tc8<d,\Ce8
436£,\O
{
Gz,\*u6-2H<8
fE*A,h.R/1*32N43698
G
:0>?6A@5*3Ijf}2
TBa1Q|2E*3IjC
<RQ|IB*3THUAV12 WX>}@B<
_7Ij476`TBa1Q|@E*
V
Q
<
>?6A@5*
8DC
G<d,<RQSIB*7THUAV12WX>Q
4£Z4-Q,G=^5_MZxTHI`4-2NabZ<6AIBG8
Q0698D>}2<6-IBG8;:YfEG8
O1Q|I
4£Z4-Q,\G=^5_]Z<
fE*A,h'I
G
r
∗
(J*-,h.7:>?6-@H8 _uIj476Q|TH2NC25*u'UjTBO;8THIvTc<2EQC
G2N/I\WXO1QSIjO1Q|IjC
6-2fE*A,h.
¥HNq?[k'l[m
n1o
B
jpNr
∗
25/I\WXO1Q|IjO1Q|I
fE*A,h.#^
*MIjf5^1,*7O1Ijf}2
_uIj436
fE*A,h.iIjC
*3Ijf5^1X,\*3O?8C
G2N/I\WXO1Q|IjO1QSI
fE*A,h.#^
/^436-Ijf}2x_uIj436fE*A,h.iIjC
/I\WXO?8C:,
G25/I\WXO1Q|IjO1Q|If}*-,h.i^
/I\WXO1Ijf}2
fE*A,h.iIjC
/^4368Cr
∗
,L>5,\FHGIjfE2fE*A,h.#^/1*M2N436-Ijf}2=:
ω
(G) = α(G)
r
∗
,L>5,\FHGIjfE2fE*A,h.#^/1*M2N436-Ijf}2=:
δ
(G) + ∆(G) = δ(G) + ∆(G) = |V | − 1
r
#" $&%
z
%D¬
«