Дискретная математика:"Графы".

Gi?(V,X)
?en. 1
Caaa?a1 Aey iai?eaioe?iaaiiiai a?aoa G, annioee?iaaiiiai n a?aoii Gi? auienaou
(ia?aioia?iaaa aa?oeiu) :
a) iii?anoai aa?oei V e iii?anoai ?aaa? X, G(V,X);
a) nienee nia?iinoe;
a) iao?eoo eioeaaioiinoe;
a) iao?eoo aania.
a) Aey a?aoa Gi? auienaou iao?eoo nia?iinoe.
Ioia?aoey aa?oei - ni. ?en 1
a) V={0,1,2,3,4,5,6,7,8,9}
X={{0,1},{0,2},{0,3},{1,2},{1,4},{1,5},{1,6},{1,7},{2,3},{2,5},{3,8},{3,9},{4,5},{4
,6},{5,3},{5,6},{5,8},{6,9},{7,8},{7,9},{8,9}}
A aaeuiaeoai ?aa?a aoaoo iaicia?aouny iiia?aie a oeacaiiii ii?yaea ia?eiay n ioey.
a) A0={1,2,3};
A1={0,2,4,5,6,7};
A2={0,1,3,5};
A3={0,2,5,8,9};
A4={1,5,6};
A5={1,2,3,4,6,8};
A6={1,4,5,9};
A7={1,8,9};
A8={1,3,5,7,9};
A9={3,6,7,8};
a) Ioia?aoey aa?oei e ?aaa? niioaaonoaaiii i. a)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
1
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
3
0
0
1
0
0
0
0
0
1
0
1
1
0
0
1
0
0
0
0
0
0
4
0
0
0
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
5
0
0
0
0
0
1
0
0
0
1
0
0
1
0
1
1
1
0
0
0
0
6
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
1
0
0
0
7
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
8
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
1
9
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
1
a) Iieacaia aa?oiyy iieiaeia iao?eou, o.e. iao?eoa aania iai?eaioe?iaaiiiai a?aoa
neiiao?e?ia ioiineoaeuii aeaaiie aeaaiiaee.
0
1
2
3
4
5
6
7
8
9
0
*
8
3
5
*
*
*
*
*
*
1
*
1
*
2
2
4
5
*
*
2
*
2
*
5
*
*
*
*
3
*
*
1
*
*
1
6
4
*
4
2
*
*
*
5
*
2
*
1
*
6
*
*
*
2
7
*
1
1
8
*
6
9
*
a) Iao?eoa nia?iinoe aey a?aoa Gi?.
0
1
2
3
4
5
6
7
8
9
0
*
1
1
1
*
*
*
*
*
*
1
-1
*
1
*
1
1
1
1
*
*
2
-1
-1
*
1
*
1
*
*
*
*
3
-1
*
-1
*
*
-1
*
*
1
1
4
*
-1
*
*
*
1
1
*
*
*
5
*
-1
-1
1
-1
*
1
*
1
*
6
*
-1
*
*
-1
-1
*
*
*
1
7
*
-1
*
*
*
*
*
*
1
1
8
*
*
*
-1
*
-1
*
-1
*
1
9
*
*
*
-1
*
*
-1
-1
-1
*
Caaa?a 2 Iaeoe aeaiao? D(G), ?aaeon R(G), eiee?anoai oaio?ia Z(G) aey a?aoa G
; oeacaou aa?oeiu, yaey?ueany oaio?aie a?aoa G.
D(G)=2
R(G)=2
Z(G)=10
Ana aa?oeiu a?aoa G(V,X) yaey?ony oaio?aie.
Caaa?a 3 Ia?aioia?iaaou aa?oeiu a?aoa G, eniieucoy aeai?eoiu:
a) "iienea a aeoaeio";
a) "iienea a oe?eio".
Enoiaiay aa?oeia - *.
a)
a)
Caaa?a 4 Eniieucoy aeai?eoi I?eia iaeoe inoia ieieiaeuiiai aana a?aoa G. auienaou
eia oeeaaee ia ieineinoe iaeaaiiiai aa?aaa, i?eiya ca ei?iaao? aa?oeio *.
Aan iaeaaiiiai aa?aaa - 14.
Eia oeeaaee aa?aaa: 000011000001111111.
Caaa?a 5 Eniieucoy aeai?eoi Aaeeno?a iaeoe aa?ai e?ao?aeoeo iooae ec aa?oeiu
* a?aoa G.
Aan iaeaaiiiai iooe - 8.
Caaa?a 6 Eniieucoy aeai?eoi Oi?aa - Oaeea?niia, iaeoe iaeneiaeuiue iioie ai
acaaoaiiie aaoiie?niie i?eaioe?iaaiiie naoe {Gi? , * , w}. Oeacaou ?ac?ac ieieiaeuiiai aana.
Iineaaiaaoaeuiinou ianuuaiey naoe (ianuuaiiua ?aa?a ioia?aiu e?o?a?eaie):
1-e oaa
2-e oaa
3-e oaa
4-e oaa
5-e oaa
6-e oaa
7-e oaa
Ieii?aoaeuii eiaai:
Eae aeaii ec ?enoiea, ?aa?a {6,9},{7,9},{3,9}, ieoa?uea aa?oeio *, ianuuaiiu, a
inoaaoaany ?aa?i {8,9}, ieoa?uaany io aa?oeiu 8, ia ii?ao iieo?eou aieuoaa cia?aiea aaniaie
ooieoee, oae eae ianuuaiiu ana ?aa?a, ieoa?uea aa?oeio 8. A?oaeie neiaaie - anee ioa?ineou
ana ianuuaiiua ?aa?a, oi aa?oeia * iaainoe?eia, ?oi yaeyaony i?eciaeii iaeneiaeuiiai iioiea a
naoe.
Iaeneiaeuiue iioie a naoe ?aaai 12.
Ieieiaeuiue ?ac?ac naoe ii ?eneo ?aaa?: {{0,1},{0,2},{0,3}}. Aai i?iioneiay
niiniaiinou ?aaia 16
Ieieiaeuiue ?ac?ac naoe ii i?iioneiie niiniaiinoe: {{6,9}, {7,9}, {3,9}, {3,8}, {5,8},
{7,8}}. Aai i?iioneiay niiniaiinou ?aaia 12.
Caaa?a 7 (Caaa?a i ii?oaeuiia) Auienaou noaiaiio? iineaaiaaoaeuiinou aa?oei
a?aoa G.
a) Oeacaou a a?aoa G Yeea?iao oaiu. Anee oaeiaie oaie ia nouanoaoao, oi a a?aoa G aiaaaeou
iaeiaiuoaa ?enei ?aaa? oaeei ia?acii, ?oiau a iiaii a?aoa ii?ii auei oeacaou Yeea?iao oaiu.
a) Oeacaou a a?aoa G Yeea?ia oeee. Anee oaeiai oeeea ia nouanoaoao, oi a a?aoa G aiaaaeou
iaeiaiuoaa ?enei ?aaa? oaeei ia?acii, ?oiau a iiaii a?aoa ii?ii auei oeacaou Yeea?ia oeee.
Noaiaiiay iineaaiaaoaeuiinou aa?oei a?aoa G:
(3,6,4,5,3,6,4,3,4,4)
a) Aey nouanoaiaaiey Yeea?iaie oaie aiionoeii oieuei aaa aa?oeiu n ia?aoiuie noaiaiyie,
iiyoiio iaiaoiaeii aiaaaeou iaii ?aa?i, nea?ai ia?ao aa?oeiaie 4 e 7.
Iieo?aiiay Yeea?iaa oaiu: 0,3,2,0,1,2,5,1,4,5,6,1,7,4,6,9,7,8,9,3,8,5,3.
Noaia Yeea?iaie oaie (aiaaaeaiiia ?aa?i iieacaii ioieoe?ii):
a) Aiaeiae?ii ioieoo a) aiaaaeyai ?aa?i {3,0}, caiueay Yeea?iao oaiu (i?e yoii auiieiyy
oneiaea nouanoaiaaiey Yeea?iaa oeeea - ?aoiinou noaiaiae anao aa?oei). ?aa?i {3,0} e?aoiia,
?oi ia i?ioeai?a?eo caaaie?, ii i?e iaiaoiaeiinoe ii?ii aaanoe ?aa?a {0,7} e {4,3} aianoi ?aiaa
aaaaaiiuo.
Iieo?aiiue Yeea?ia oeee: 0,3,2,0,1,2,5,1,4,5,6,1,7,4,6,9,7,8,9,3,8,5,3,0.
Noaia Yeea?iaa oeeea (aiaaaeaiiua ?aa?a iieacaiu ioieoe?ii):
Caaa?a 8
a) Oeacaou a a?aoa Gi? Aaieeuoiiia ioou. Anee oaeie ioou ia nouanoaoao, oi a a?aoa Gi?
eciaieou i?eaioaoe? iaeiaiuoaai ?enea ?aaa? oaeei ia?acii, ?oiau a iiaii a?aoa Aaieeuoiiia ioou
ii?ii auei oeacaou.
a) Oeacaou a a?aoa Gi? Aaieeuoiiia oeee. Anee oaeie oeee ia nouanoaoao, oi a a?aoa Gi?
eciaieou i?eaioaoe? iaeiaiuoaai ?enea ?aaa? oaeei ia?acii, ?oiau a iiaii a?aoa Aaieeuoiiia oeee
ii?ii auei oeacaou.
a) Aaieeuoiiia ioou (?aa?a n eciaiaiiie i?eaioaoeae iieacaiu ioieoe?ii):
a) Aaieeuoiiia oeee (?aa?a n eciaiaiiie i?eaioaoeae iieacaiu ioieoe?ii):
Caaa?a 9 (Caaa?a i eiiieaiy?a?a) Aai iieiue i?eaioe?iaaiiue neiiao?e?aneee a?ao
n aa?oeiaie x1, x2,...xn.Aan aoae xixj caaai yeaiaioaie Vij iao?eou aania. Eniieucoy aeai?eoi
iaoiaa aaoaae e a?aieo, iaeoe Aaieeuoiiia eiioo? ieieiaeuiiai (iaeneiaeuiiai) aana. Caaa?o ia
iaeneiaeuiia cia?aiea Aaieeuoiiiaa eiioo?a naanoe e caaa?a ia ieieiaeuiia cia?aiea, ?anniio?aa
iao?eoo n yeaiaioaie ,aaa . Auiieieou ?enoiie.
Enoiaiay oaaeeoa.
x1
x2
x3
x4
x5
x6
x1
*
3
7
2
*
11
x2
8
*
06
*
4
3
x3
6
05
*
7
*
2
x4
6
*
13
*
5
*
x5
3
3
3
4
*
5
x6
8
6
*
2
2
*
Oaaeeoa A 14
x1
x2
x3
x4
x5
x6
x1
*
1
5
01
*
7
2
x2
8
*
01
*
4
1
x3
6
00
*
7
*
00
x4
1
*
8
*
01
*
5
x5
01
00
00
1
*
00
3
x6
6
4
*
00
00
*
2
2
A?iaei ii ia?aoiao x2-x3:
Oaaeeoa 23 *=14+0=14
x1
x2
x4
x5
x6
x1
*
1
01
*
7
x3
6
*
7
*
06
x4
1
*
*
01
*
x5
01
01
1
*
00
x6
6
4
00
00
*
Oaaeeoa 23 *=14+1=15
x1
x2
x3
x4
x5
x6
x1
*
1
5
01
*
7
x2
7
*
*
*
3
03
1
x3
6
00
*
7
*
00
x4
1
*
8
*
01
*
x5
01
00
05
1
*
00
x6
6
4
*
00
00
*
I?iaie?aai ii 23. A?iaei ii ia?aoiao x3-x6:
Oaaeeoa 23E36 *=14+0=14
x1
x2
x4
x5
x1
*
1
01
*
x4
1
*
*
01
x5
01
01
1
*
x6
6
*
00
00
Oaaeeoa 23 36 *=14+6=20
x1
x2
x4
x5
x6
x1
*
1
01
*
7
x3
01
*
1
*
*
6
x4
1
*
*
01
*
x5
00
01
1
*
07
x6
6
4
00
00
*
I?iaie?aai ii 23 36. A?iaei ii ia?aoiao x4-x5:
Oaaeeoa 23E36 45 *=14+0=14
x1
x2
x4
x1
*
1
01
x5
01
01
1
x6
6
*
00
Oaaeeoa 23 36 45 *=14+1=15
x1
x2
x4
x5
x1
*
1
01
*
x4
00
*
*
*
1
x5
01
01
1
*
x6
6
*
00
00
I?iaie?aai ii 23 36 45. A?iaei ii ia?aoiao x5-x1:
Oaaeeoa 23 36 45 51 *=14+1=15
x2
x4
x1
1
*
1
x6
*
00
Oaaeeoa 23 36 45 51 *=14+6=20
x1
x2
x4
x1
*
1
01
x5
*
01
*
x6
0
*
00
6
Ieii?aoaeuii eiaai Aaieeuoiiia eiioo?: 2,3,6,4,5,1,2.
I?aaa?aai ?acaeaiee:
Caaa?a 10 (Caaa?a i iacia?aieyo) Aai iieiue aaoaieuiue a?ao Knn n aa?oeiaie ia?aie
aiee x1, x2,...xn.e aa?oeiaie a?oaie aiee y1, y2,...yn..Aan ?aa?a {xi,yj} caaaaony yeaiaioaie vij
iao?eou aania. Eniieucoy aaiaa?neee aeai?eoi, iaeoe niaa?oaiiia ia?ini?aoaiea ieieiaeuiiai
(iaeneiaeuiiai aana). Auiieieou ?enoiie.
Iao?eoa aania aaoaieuiiai a?aoa K55 :
y1
y2
y3
y4
y5
x1
2
0
0
0
0
x2
0
7
9
8
6
x3
0
1
3
2
2
x4
0
8
7
6
4
x5
0
7
6
8
3
Ia?aue yoai - iieo?aiea ioeae ia io?ai, o. e. ioee o?a anou ai anao no?ie e noieaoao.
Aoi?ie yoai - iaoi?aaiea iieiiai ia?ini?aoaiey.
y1
y2
y3
y4
y5
x1
2
0
0
0
0
x2
0
7
9
8
6
x3
0
1
3
2
2
x4
0
8
7
6
4
x5
0
7
6
8
3
O?aoee yoai - iaoi?aaiea iaeneiaeuiiai ia?ini?aoaiey.
y1
y2
y3
y4
y5
x1
2
0
0
0
0
X
x2
0
7
9
8
6
X
x3
0
1
3
2
2
x4
0
8
7
6
4
x5
0
7
6
8
3
X
X
?aoaa?oue yoai - iaoi?aaiea ieieiaeuiie iii?u.
y1
y2
y3
y4
y5
x1
2
0
0
0
0
x2
0
7
9
8
6
5
x3
0
1
3
2
2
1
x4
0
8
7
6
4
2
x5
0
7
6
8
3
3
4
Iyoue yoai - aicii?iay ia?anoaiiaea iaeioi?uo ioeae.
y1
y2
y3
y4
y5
x1
3
0
0
0
0
x2
0
6
8
7
5
5
x3
0
0
2
1
1
1
x4
0
7
6
5
3
2
x5
0
6
5
7
2
3
4
?aoaiea n iaioeaaui cia?aieai. Ia?aoia ei aoi?iio yoaio.
Iieiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
x2
0
6
8
7
5
x3
0
0
2
1
1
x4
0
7
6
5
3
x5
0
6
5
7
2
Iaeneiaeuiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
X
x2
0
6
8
7
5
X
x3
0
0
2
1
1
x4
0
7
6
5
3
x5
0
6
5
7
2
X
X
Ieieiaeuiay iii?a:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
6
x2
0
6
8
7
5
7
x3
0
0
2
1
1
1
x4
0
7
6
5
3
2
x5
0
6
5
7
2
3
4
5
Ia?anoaiiaea ioeae:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
6
x2
0
6
8
7
5
7
x3
0
0
2
1
1
1
x4
0
7
6
5
3
2
x5
0
6
5
7
2
3
4
5
Iieiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
6
x2
0
6
8
7
5
7
x3
0
0
2
1
1
1
x4
0
7
6
5
3
2
x5
0
6
5
7
2
3
4
5
Iaeneiaeuiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
X
x2
0
6
8
7
5
x3
0
0
2
1
1
X
x4
0
7
6
5
3
X
x5
0
6
5
7
2
X
X
X
Ieieiaeuiay iii?a:
y1
y2
y3
y4
y5
x1
3
0
0
0
0
x2
0
6
8
7
5
1
x3
0
0
2
1
1
x4
0
7
6
5
3
x5
0
6
5
7
2
2
3
Ia?anoaiiaea ioeae:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
x2
0
4
6
5
3
1
x3
2
0
2
1
1
x4
2
7
6
5
3
x5
0
4
3
5
0
2
3
Iieiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
x2
0
4
6
5
3
x3
2
0
2
1
1
x4
2
7
6
5
3
x5
0
4
3
5
0
Iaeneiaeuiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
X
x2
0
4
6
5
3
X
x3
2
0
2
1
1
X
x4
2
7
6
5
3
x5
0
4
3
5
0
X
X
X
X
X
Ieieiaeuiay iii?a:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
x2
0
4
6
5
3
x3
2
0
2
1
1
x4
2
7
6
5
3
1
x5
0
4
3
5
0
Ia?anoaiiaea ioeae:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
x2
0
4
6
5
3
x3
2
0
2
1
1
x4
0
5
4
3
1
1
x5
0
4
3
5
0
Iieiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
x2
0
4
6
5
3
x3
2
0
2
1
1
x4
0
5
4
3
1
1
x5
0
4
3
5
0
Iaeneiaeuiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
X
x2
0
4
6
5
3
X
x3
2
0
2
1
1
X
x4
0
5
4
3
1
x5
0
4
3
5
0
X
X
X
X
X
Ieieiaeuiay iii?a:
y1
y2
y3
y4
y5
x1
5
0
0
0
0
x2
0
4
6
5
3
3
x3
2
0
2
1
1
x4
0
5
4
3
1
1
x5
0
4
3
5
0
2
Ia?anoaiiaea ioeae:
y1
y2
y3
y4
y5
x1
6
0
0
0
0
x2
0
3
5
4
2
3
x3
3
0
2
1
1
x4
0
4
3
2
0
1
x5
1
4
3
5
0
2
Iieiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
6
0
0
0
0
x2
0
3
5
4
2
3
x3
3
0
2
1
1
x4
0
4
3
2
0
1
x5
1
4
3
5
0
2
Iaeneiaeuiia ia?ini?aoaiea:
y1
y2
y3
y4
y5
x1
6
0
0
0
0
X
x2
0
3
5
4
2
X
x3
3
0
2
1
1
X
x4
0
4
3
2
0
x5
1
4
3
5
0
X
X
X
X
X
Ieieiaeuiay iii?a:
y1
y2
y3
y4
y5
x1
6
0
0
0
0
x2
0
3
5
4
2
4
x3
3
0
2
1
1
x4
0
4
3
2
0
1
x5
1
4
3
5
0
5
2
3
A ?acoeuoaoa eiaai:
y1
y2
y3
y4
y5
x1
6
0
0
0
0
x2
0
1
3
2
2
4
x3
3
0
2
1
1
x4
0
2
1
0
0
1
x5
1
4
3
5
0
5
2
3
Enoiaiue a?ao
Iieo?aiiue a?ao:
Aan iaeaaiiiai niaa?oaiiiai ia?ini?aoaiey = 12.
Caaa?a 11 ?aoeou caaa?o 10, eniieucoy aeai?eoi aaoaae e a?aieo (ioi?aanoaea
aa?oeiu xi e yj).
Oaaeeoa A (enoiaiay). No?iee - xi , noieaou - yj. *=0
1
2
3
4
5
1
2
01
03
02
02
2
06
7
9
8
6
3
01
1
3
2
2
4
04
8
7
6
4
5
03
7
6
8
3
A?iaei ii ia?aoiao x2 - y1:
Oaaeeoa A21 *=0+8=8
2
3
4
5
1
00
02
01
00
3
01
2
1
1
1
4
4
3
2
02
4
5
4
3
5
03
3
Oaaeeoa 21 *=0+6=6
1
2
3
4
5
1
2
01
03
02
00
2
*
1
3
2
01
6
3
01
1
3
2
2
4
04
8
7
6
4
5
03
7
6
8
3
I?iaie?aai ii 21:
A?iaei ii ia?aoiao x4 - y1:
Oaaeeoa 21A41 *=6+4=10
2
3
4
5
1
00
02
01
00
2
1
3
2
01
3
01
2
1
1
1
5
4
3
5
03
3
Oaaeeoa 21 41 *=6+4=10
1
2
3
4
5
1
2
01
03
02
00
2
*
1
3
2
01
3
01
1
3
2
2
4
*
4
3
2
02
4
5
03
7
6
8
3
I?iaie?aai ii A21:
A?iaei ii ia?aoiao x5 - y5:
Oaaeeoa A21A55 *=8+2=10
2
3
4
1
00
01
00
3
01
2
1
4
2
1
01
2
Oaaeeoa A21 55 *=8+3=11
2
3
4
5
1
00
02
01
00
3
01
2
1
1
4
4
3
2
02
5
1
01
2
*
3
I?iaie?aai ii A21A55:
A?iaei ii ia?aoiao x3 - y2:
Oaaeeoa A21A55A32 *=10+0=10
3
4
1
01
00
4
1
01
Aaeaa ?aoaiea i?aaeaii: x1 - y3 e x4 - y4. Yoi ia oaaee?eo ioaieo.
A eoiaa eiaai niaa?oaiiia ia?ini?aoaiea n ieieiaeuiui aanii:
I?aaa?aai ?acaeaiee: