LTDT Chuong 1

L THUYT [email protected]

GV: Dng Anh c

NI DUNGi cng v thCyCc bi ton ng i th phng v bi ton t mu thMng v bi ton lung trn mng, bi ton cp ghpGV: Dng Anh c*

GV: Dng Anh c

TI LIU THAM KHOGio trnh L Thuyt Th - Dng Anh c, Trn an ThTon ri rc Nguyn T Thnh, Nguyn c Ngha...GV: Dng Anh c*

GV: Dng Anh c

I CNG V TH

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NH NGHAMt th c hng G=(X, U) c nh ngha bi:Tp hp X c gi l tp cc nh ca th;Tp hp U l tp cc cnh ca th;Mi cnh uU c lin kt vi mt cp nh (i, j)X2.GV: Dng Anh c*

GV: Dng Anh c

NH NGHAMt th v hng G=(X, E) c nh ngha bi:Tp hp X c gi l tp cc nh ca th;Tp hp E l tp cc cnh ca th;Mi cnh eE c lin kt vi mt cp nh {i, j}X2, khng phn bit th tGV: Dng Anh c*

GV: Dng Anh c

TH HU HN th c tp nh v tp cnh hu hn c gi l TH HU HNHc phn ny ch lm vic cc TH HU HN, tuy nhin ngn gn chng ta ch dng thut ng TH v hiu ngm l th hu hn.GV: Dng Anh c*

GV: Dng Anh c

NH KTrn th c hng, xt cnh u c lin kt vi cp nh (i, j):

Cnh u k vi nh i v nh j (hay nh i v nh j k vi cnh u); c th vit tt u=(i, j). Cnh u i ra khi nh i v i vo nh jnh j c gi l nh k ca nh iGV: Dng Anh c*

GV: Dng Anh c

NH KTrn th v hng, xt cnh e c lin kt vi cp nh (i, j):

Cnh e k vi nh i v nh j (hay nh i v nh j k vi cnh e); c th vit tt e=(i, j). nh i v nh j c gi l 2 nh k nhau (hay nh i k vi nh j v ngc li, nh j k vi nh i)

GV: Dng Anh c*

GV: Dng Anh c

MT S KHI NIMCnh song songKhuynnh treonh c lpGV: Dng Anh c*

GV: Dng Anh c

CC DNG TH th RNG: tp cnh l tp rng th N: khng c khuyn v cnh song song th : th v hng, n, gia hai nh bt k u c ng mt cnh. th N nh k hiu l KN.KN c N(N-1)/2 cnh.GV: Dng Anh c*CAB

GV: Dng Anh c

CC DNG TH th LNG PHN: th G=(X, E) c gi l th lng phn nu tp X c chia thnh hai tp X1 v X2 tha: X1 v X2 phn hoch X;Cnh ch ni gia X1 v X2. th LNG PHN : l th lng phn n, v hng tha vi (i, j)/iX1 v jX2 c ng mt cnh i v j.X1=N v X2=M, k hiu KM, N.GV: Dng Anh c*CABDE

GV: Dng Anh c

V D: TH GV: Dng Anh c*

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BC CA NHXt th v hng GBc ca nh x trong th G l s cc cnh k vi nh x, mi khuyn c tnh hai ln, k hiu l dG(x) (hay d(x) nu ang xt mt th no ).GV: Dng Anh c*

GV: Dng Anh c

BC CA THXt th c hng GNa bc ngoi ca nh x l s cc cnh i ra khi nh x, k hiu d+(x).Na bc trong ca nh x l s cc cnh i vo nh x, k hiu d-(x).Bc ca nh x: d(x)=d+(x)+d-(x)GV: Dng Anh c*

GV: Dng Anh c

BC CA NHnh TREO l nh c bc bng 1.nh C LP l nh c bc bng 0.GV: Dng Anh c*CABD

GV: Dng Anh c

MI LIN H BC - S CNHnh l:Xt th c hng G=(X, U). Ta c:

Xt th v hng G=(X, E). Ta c:

H qu: s lng cc nh c bc l trong mt th l mt s chn.GV: Dng Anh c*

GV: Dng Anh c

NG CU THHai th v hng G1 =(X1, E1) v G2=(X2, E2) c gi l ng cu vi nhau nu tn ti hai song nh v tha mn iu kin:: X1 X2 v : E1 E2Nu cnh e E1 k vi cp nh {x, y} X1 trong G1 th cnh (e) s k vi cp nh {(x), (y)} trong G2 (s tng ng cnh).GV: Dng Anh c*G1G2

GV: Dng Anh c

NG CU THHai th c hng G1=(X1, U1) v G2=(X2, U2) c gi l ng cu vi nhau nu tn ti hai song nh v tha mn iu kin:: X1 X2 v : U1 U2Nu cnh u U1 lin kt vi cp nh (x, y) X1 trong G1 th cnh (u) s lin kt vi cp nh ((x), (y)) trong G2 (s tng ng cnh).GV: Dng Anh c*

GV: Dng Anh c

TH CONXt hai th G=(X, U) v G1=(X1, U1). G1 c gi l th con ca G v k hiu G1 G nu:X1 X; U1 Uu=(i, j) U ca G, nu u U1 th i, j X1GV: Dng Anh c*GG1

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TH B PHN th con G1=(X1, U1) ca th G=(X, U) c gi l th b phn ca G nu X=X1.GV: Dng Anh c*GG1

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TH CON SINH BI TP NHCho th G=(X, U) v A X. th con sinh bi tp nh A, k hiu (A, V), trong :(i) tp cnh V U(ii) Gi u=(i, j) U l mt cnh ca G, nu i, j A th u VGV: Dng Anh c*G

A={1, 2, 4}

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DY CHUYN, CHU TRNHMt dy chuyn trong G=(X, U) l mt th con C=(V, E) ca G vi:V = {x1, x2, , xM}E = {u1, u2, , uM-1} vi u1=x1x2, u2=x2x3, , uM-1=xM-1xM; lin kt xixi+1 khng phn bit th t.Khi , x1 v xM c ni vi nhau bng dy chuyn C. x1 l nh u v xM l nh cui ca C.S cnh ca C c gi l di ca C.Khi cc cnh hon ton xc nh bi cp nh k, dy chuyn c th vit gn (x1, x2, , xM)GV: Dng Anh c*

GV: Dng Anh c

DY CHUYN, CHU TRNHDy chuyn S CP: dy chuyn khng c nh lp li.CHU TRNH: l mt dy chuyn c nh u v nh cui trng nhau. GV: Dng Anh c*

GV: Dng Anh c

NG I, MCHMt NG I trong G=(X, U) l mt th con P=(V, E) ca G vi:V = {x1, x2, , xM}E = {u1, u2, , uM-1} vi u1=x1x2, u2=x2x3, , uM-1=xM-1xM; lin kt xixi+1 theo ng th t.Khi , c ng i P ni t x1 n xM. x1 l nh u v xM l nh cui ca P.S cnh ca P c gi l di ca P.Khi cc cnh hon ton xc nh bi cp nh k, ng i c th vit gn (x1, x2, , xM)GV: Dng Anh c*

GV: Dng Anh c

ng i S CP: ng i khng c nh lp li.MCH: l mt ng i c nh u trng vi nh cuiVi th v hng:Dy chuyn ng i, chu trnh mch.Do , thut ng ng i cng c dng cho th v hng. Mch trong th c hng cn c gi l chu trnh c hng. ng i trong th c hng cng c gi l ng i c hng nhn mnh.

GV: Dng Anh c*NG I, MCH

GV: Dng Anh c

THNH PHN LIN THNGCho th G=(X, U). Ta nh ngha mt quan h LIN KT nh sau trn tp nh X:i, jX, i j (ij hoc c dy chuyn ni i vi j).Quan h ny c ba tnh cht: phn x, i xng v bc cu nn n l mt quan h tng ng. Do tp X c phn hoch thnh cc lp tng ng.

GV: Dng Anh c*

GV: Dng Anh c

THNH PHN LIN THNGnh ngha:Mt thnh phn lin thng ca th l mt lp tng ng c xc nh bi quan h LIN KT ;S thnh phn lin thng ca th l s lng cc lp tng ng; th lin thng l th ch c mt thnh phn lin thng.Khi mt G gm p thnh phn lin thng G1, G2, , Gp th cc th Gi cng l cc th con ca G v dG(x) = dGi(x), x ca Gi.

L thuyt th - Nguyn Thanh Sn*

GV: Dng Anh c

THNH PHN LIN THNGG gm 2 thnh phn lin thng, H l th lin thngGV: Dng Anh c*GH

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THNH PHN LIN THNGThut ton xc nh cc thnh phn lin thngInput: th G=(X, E), tp X gm N nh 1, 2, , NOutput: cc nh ca G c gn nhn l s hiu ca thnh phn lin thng tng ngKhi to bin label=0 v gn nhn 0 cho tt c cc nhDuyt qua tt c cc nh iXNu nhn ca i l 0label = label + 1Gn nhn cho tt c cc nh cng thuc thnh phn lin thng vi i l label

GV: Dng Anh c*

GV: Dng Anh c

THNH PHN LIN THNGThut ton gn nhn cc nh cng thuc thnh phn lin thng vi nh i Visit(i, label)Input: th G=(X, E), nh i, nhn labelOutput: cc nh cng thuc thnh phn lin thng vi i c gn nhn labelGn nhn label cho nh iDuyt qua tt c cc nh jX v c cnh ni vi iNu nhn ca j l 0Visit(j, label)GV: Dng Anh c*

GV: Dng Anh c

BIU DIN TH BNG HNH VL thuyt th - Nguyn Thanh Sn*ABCDu1u2u3u4u5u6ABCDe1e2e3e4e5e6GH

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BIU DIN TH BNG MA TRNMa trn K:Xt th G=(X, U), gi s tp X gm N nh v c sp th t X={x1, x2, , xN}, tp U gm M cnh v c sp th t U={u1, u2, , uM}.Ma trn k ca th G, k hiu B(G), l mt ma trn nh phn cp NxN B=(Bij) vi Bij c nh ngha:Bij=1 nu c cnh ni xi ti xj, Bij=0 trong trng hp ngc li.


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BIU DIN THBNG MA TRN KL thuyt th - Nguyn Thanh Sn*1234G

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BIU DIN TH BNG MA TRN KL thuyt th - Nguyn Thanh Sn*1234G

GV: Dng Anh c

BIU DIN TH BNG MA TRNMa trn LIN THUC ca th v hng:Xt th G=(X, U) v hng, gi s tp X gm N nh v c sp th t X={x1, x2, , xN}, tp U gm M cnh v c sp th t U={u1, u2, , uM}.Ma trn lin thuc (hay lin kt nh cnh) ca G, k hiu A(G), l ma trn nh phn cp NxM A=(Aij) vi Aij c nh ngha:Aij=1 nu nh xi k vi cnh uj, Aij=0 nu ngc li.


GV: Dng Anh c

BIU DIN TH BNG MA TRN LIN THUCL thuyt th - Nguyn Thanh Sn*G1234e1e2e3e4e5e6

GV: Dng Anh c

BIU DIN TH BNG MA TRNMa trn LIN THUC ca th c hng:Xt th G=(X, U) c hng, gi s tp X gm N nh v c sp th t X={x1, x2, , xN}, tp U gm M cnh v c sp th t U={u1, u2, , uM}.Ma trn lin thuc (hay lin kt nh cnh) ca G, k hiu A(G), l ma trn nh phn cp NxM A=(Aij) vi Aij c nh ngha:Aij=1 nu cnh uj i ra khi nh xi, Aij=-1 nu cnh uj i vo nh xi,Aij=0 trong cc trng hp khc.


GV: Dng Anh c

BIU DIN TH BNG MA TRN LIN THUCL thuyt th - Nguyn Thanh Sn*G1234u1u2u3u4u5u6

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BIU DIN TH BNG NNLT C++#define MAX 100class Graph{protected:int nVertex; //s nh ca th, cc nh c //nh s t 0int labels[MAX]; //nhn ca cc nhint degrees[MAX];//bc cc nhunsigned char B[MAX][MAX]; //ma trn kvoid Visit(int i, int label);public:void GetData(const char *filename);int FindConnected();}L thuyt th - Nguyn Thanh Sn*

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Source code: nhp d liu t textfilevoid Graph::GetData(const char *filename){//nhp d liu t tp tin vn bnifstream fin;fin.open(filename);fin >> nVertex;for (int i = 0; i < nVertex; ++i)for (int j = 0; j < nVertex; ++j)fin >> B[i][j];fin.close();}


GV: Dng Anh c

Source code: xc nh bc ca nhvoid Graph::CountDegree(){//xc nh bc ca cc nh, th v hngfor(int i=0;i

Source code: gn nhn 1 TPLTvoid Graph::Visit(int i, int label){labels[i] = label;for (int j=0; j

Source code: gn nhn tt c TPLTint Graph::FindConnected(){int i, label;for (int i=0; i

BI TPG l mt th n, v hng c s nh N>3. Chng minh G c cha 2 nh cng bc. th G c ng 2 nh bc l. Chng minh tn ti mt dy chuyn ni hai nh vi nhau.Xt th G n, v hng gm N nh, M cnh v P thnh phn lin thng.Chng minh: M (N-P)(N-P+1)/2,suy ra nu M > (N-1)(N-2)/2 th G lin thng.Mt th n c 10 nh, 37 cnh th c chc lin thng hay khng?

GV: Dng Anh c*

GV: Dng Anh c

BI TP th G n, v hng gm N nh v d(x)(N-1)/2 vi mi nh x. Chng minh G lin thng. th v hng G lin thng gm N nh. Chng minh s cnh ca G N-1.Xt th G v hng n. Gi x l nh c bc nh nht ca G. Gi s d(x)k2 vi k nguyn dng. Chng minh G cha mt chu trnh s cp c chiu di ln hn hay bng k+1.GV: Dng Anh c*

GV: Dng Anh c

BI TPCho G l th v hng lin thng. Gi s C1 v C2 l 2 dy chuyn s cp trong G c s cnh nhiu nht. Chng minh C1 v C2 c nh chung.G l th v hng khng khuyn v d(x) 3 vi mi nh x. Chng minh G c cha chu trnh vi s cnh chn.GV: Dng Anh c*

GV: Dng Anh c

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LTDT Chuong 1