 Giáo trình ĐẠI CƯƠNG ĐỒ THỊ Mu . c lu . c L`o . i n´oi d¯ˆa ` u 7 1 D - a . i cu . o . ng vˆe ` d¯ˆo ` thi . 9 1.1 D - i . nh ngh˜ıa v`a c´ac kh´ai niˆe . m . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 D - ˆo ` thi . c´o hu . ´o . ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 D - ˆo ` thi . v`a ´anh xa . d¯a tri . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.3 D - ˆo ` thi . vˆo hu . ´o . ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 C´ac d¯i . nh ngh˜ıa ch´ınh . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Ma trˆa . n biˆe ˙’ u diˆe ˜ n d¯ˆo ` thi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 Ma trˆa . n liˆen thuˆo . c d¯ı ˙’ nh-cung . . . . . . . . . . . . . . . . . . . . . . 13 1.2.2 Ma trˆa . n liˆen thuˆo . c d¯ı ˙’ nh-ca . nh . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Ma trˆa . n kˆe ` hay ma trˆa . n liˆen thuˆo . c d¯ı ˙’ nh-d¯ı ˙’ nh . . . . . . . . . . . . . 17 1.2.4 C´ac biˆe ˙’ u diˆe ˜ n cu ˙’ a d¯ˆo ` thi . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3 T´ınh liˆen thˆong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1 Dˆay chuyˆe ` n v`a chu tr`ınh . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.2 D - u . `o . ng d¯i v`a ma . ch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.3 T´ınh liˆen thˆong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1 1.3.4 Cˆa ` u, k−liˆen thˆong . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.5 D - ˆo ` thi . liˆen thˆong ma . nh . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4 Pha . m vi v`a liˆen thˆong ma . nh . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.1 Ma trˆa . n pha . m vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.2 T`ım c´ac th`anh phˆa ` n liˆen thˆong ma . nh . . . . . . . . . . . . . . . . . . 36 1.4.3 Co . so . ˙’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.5 D - ˇa ˙’ ng cˆa ´ u cu ˙’ a c´ac d¯ˆo ` thi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.5.1 1−d¯ˇa ˙’ ng cˆa ´ u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.5.2 2−d¯ˇa ˙’ ng cˆa ´ u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.6 C´ac d¯ˆo ` thi . d¯ˇa . c biˆe . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.6.1 D - ˆo ` thi . khˆong c´o ma . ch . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.6.2 D - ˆo ` thi . phˇa ˙’ ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 C´ac sˆo ´ co . ba ˙’ n cu ˙’ a d¯ˆo ` thi . 49 2.1 Chu sˆo ´ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Sˇa ´ c sˆo ´ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2.1 C´ach t`ım sˇa ´ c sˆo ´ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3 Sˆo ´ ˆo ˙’ n d¯i . nh trong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4 Sˆo ´ ˆo ˙’ n d¯i . nh ngo`ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.5 Phu ˙’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6 Nhˆan cu ˙’ a d¯ˆo ` thi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.6.1 C´ac d¯i . nh l´y vˆe ` tˆo ` n ta . i v`a duy nhˆa ´ t . . . . . . . . . . . . . . . . . . . 69 2.6.2 Tr`o cho . i Nim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2 3 C´ac b`ai to´an vˆe ` d¯u . `o . ng d¯i 75 3.1 D - u . `o . ng d¯i gi˜u . a hai d¯ı ˙’ nh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1.1 D - u . `o . ng d¯i gi˜u . a hai d¯ı ˙’ nh . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1.2 D - ˆo ` thi . liˆen thˆong ma . nh . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 D - u . `o . ng d¯i ngˇa ´ n nhˆa ´ t gi˜u . a hai d¯ı ˙’ nh . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.1 Tru . `o . ng ho . . p ma trˆa . n tro . ng lu . o . . ng khˆong ˆam . . . . . . . . . . . . . . 78 3.2.2 Tru . `o . ng ho . . p ma trˆa . n tro . ng lu . o . . ng tu`y ´y . . . . . . . . . . . . . . . . . 82 3.3 D - u . `o . ng d¯i ngˇa ´ n nhˆa ´ t gi˜u . a tˆa ´ t ca ˙’ c´ac cˇa . p d¯ı ˙’ nh . . . . . . . . . . . . . . . . . 87 3.3.1 Thuˆa . t to´an Hedetniemi (tru . `o . ng ho . . p ma trˆa . n tro . ng lu . o . . ng khˆong ˆam) 88 3.3.2 Thuˆa . t to´an Floyd (tru . `o . ng ho . . p ma trˆa . n tro . ng lu . o . . ng tu`y ´y) . . . . . . 93 3.4 Ph´at hiˆe . n ma . ch c´o d¯ˆo . d`ai ˆam . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.1 Ma . ch tˆo ´ i u . u trong d¯ˆo ` thi . c´o hai tro . ng lu . o . . ng . . . . . . . . . . . . . . 96 4 C ˆ AY 99 4.1 Mo . ˙’ d¯ˆa ` u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Cˆay Huffman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.1 C´ac bˆo . m˜a “tˆo ´ t” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.2 M˜a Huffman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3 Cˆay bao tr`um . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.1 Thuˆa . t to´an t`ım kiˆe ´ m theo chiˆe ` u rˆo . ng x´ac d¯i . nh cˆay bao tr`um . . . . . 107 4.3.2 Thuˆa . t to´an t`ım kiˆe ´ m theo chiˆe ` u sˆau x´ac d¯i . nh cˆay bao tr`um . . . . . 107 4.3.3 T`ım cˆay bao tr`um du . . a trˆen hai ma ˙’ ng tuyˆe ´ n t´ınh . . . . . . . . . . . 108 4.3.4 Thuˆa . t to´an t`ım tˆa ´ t ca ˙’ c´ac cˆay bao tr`um . . . . . . . . . . . . . . . . 112 4.3.5 Hˆe . co . so . ˙’ cu ˙’ a c´ac chu tr`ınh d¯ˆo . c lˆa . p . . . . . . . . . . . . . . . . . . . 112 3 4.4 Cˆay bao tr`um tˆo ´ i thiˆe ˙’ u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4.1 Thuˆa . t to´an Kruskal . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.4.2 Thuˆa . t to´an Prim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.4.3 Thuˆa . t to´an Dijkstra-Kevin-Whitney . . . . . . . . . . . . . . . . . . 121 4.5 B`ai to´an Steiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5 B`ai to´an Euler v`a b`ai to´an Hamilton 127 5.1 B`ai to´an Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.1.1 Thuˆa . t to´an t`ım dˆay chuyˆe ` n Euler . . . . . . . . . . . . . . . . . . . . 129 5.2 B`ai to´an ngu . `o . i d¯u . a thu . Trung Hoa . . . . . . . . . . . . . . . . . . . . . . . 131 5.3 B`ai to´an Hamilton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3.1 C´ac d¯iˆe ` u kiˆe . n cˆa ` n d¯ˆe ˙’ tˆo ` n ta . i chu tr`ınh Hamilton . . . . . . . . . . . . 138 5.3.2 C´ac d¯iˆe ` u kiˆe . n d¯u ˙’ vˆe ` su . . tˆo ` n ta . i chu tr`ınh Hamilton . . . . . . . . . . 139 5.3.3 C´ac d¯iˆe ` u kiˆe . n d¯u ˙’ vˆe ` su . . tˆo ` n ta . i ma . ch Hamilton . . . . . . . . . . . . . 142 6 D - ˆo ` thi . phˇa ˙’ ng 149 6.1 D - i . nh ngh˜ıa v`a c´ac v´ı du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.2 C´ac biˆe ˙’ u diˆe ˜ n kh´ac nhau cu ˙’ a mˆo . t d¯ˆo ` thi . phˇa ˙’ ng . . . . . . . . . . . . . . . . 151 6.3 C´ac t´ınh chˆa ´ t cu ˙’ a d¯ˆo ` thi . phˇa ˙’ ng . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4 Ph´at hiˆe . n t´ınh phˇa ˙’ ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.4.1 Kiˆe ˙’ m tra t´ınh phˇa ˙’ ng . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.5 D - ˆo ´ i ngˆa ˜ u h`ınh ho . c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.6 D - ˆo ´ i ngˆa ˜ u tˆo ˙’ ho . . p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7 Ma . ng vˆa . n ta ˙’ i 173 4 7.1 Mo . ˙’ d¯ˆa ` u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.2 B`ai to´an luˆo ` ng l´o . n nhˆa ´ t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2.1 Thuˆa . t to´an g´an nh˜an d¯ˆe ˙’ t`ım luˆo ` ng l´o . n nhˆa ´ t . . . . . . . . . . . . . . 180 7.2.2 D - ˆo ` thi . d¯iˆe ` u chı ˙’ nh luˆo ` ng . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2.3 Phˆan t´ıch luˆo ` ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.3 C´ac ca ˙’ i biˆen d¯o . n gia ˙’ n cu ˙’ a b`ai to´an luˆo ` ng l´o . n nhˆa ´ t . . . . . . . . . . . . . . . 183 7.3.1 C´ac d¯ˆo ` thi . c´o nhiˆe ` u nguˆo ` n v`a nhiˆe ` u d¯´ıch . . . . . . . . . . . . . . . . 183 7.3.2 C´ac d¯ˆo ` thi . v´o . i r`ang buˆo . c ta . i c´ac cung v`a d¯ı ˙’ nh . . . . . . . . . . . . . 184 7.3.3 C´ac d¯ˆo ` thi . c´o cˆa . n trˆen v`a cˆa . n du . ´o . i vˆe ` luˆo ` ng . . . . . . . . . . . . . . 185 7.4 Luˆo ` ng v´o . i chi ph´ı nho ˙’ nhˆa ´ t . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.4.1 Thuˆa . t to´an Klein, Busacker, Gowen . . . . . . . . . . . . . . . . . . . 186 7.5 Cˇa . p gh´ep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.5.1 C´ac b`ai to´an vˆe ` cˇa . p gh´ep . . . . . . . . . . . . . . . . . . . . . . . . 189 7.5.2 Cˇa . p gh´ep l´o . n nhˆa ´ t trong d¯ˆo ` thi . hai phˆa ` n . . . . . . . . . . . . . . . . 192 7.5.3 Cˇa . p gh´ep ho`an ha ˙’ o trong d¯ˆo ` thi . hai phˆa ` n . . . . . . . . . . . . . . . 193 A Thu . viˆe . n Graph.h 197 T`ai liˆe . u tham kha ˙’ o 209 5 6 L`o . i n´oi d¯ˆa ` u Trong thu . . c tˆe ´ d¯ˆe ˙’ miˆeu ta ˙’ mˆo . t sˆo ´ t`ınh huˆo ´ ng ngu . `o . i ta thu . `o . ng biˆe ˙’ u thi . bˇa ` ng mˆo . t h`ınh a ˙’ nh gˆo ` m c´ac d¯iˆe ˙’ m (c´ac d¯ı ˙’ nh)-biˆe ˙’ u diˆe ˜ n c´ac thu . . c thˆe ˙’ -v`a v˜e c´ac d¯oa . n thˇa ˙’ ng nˆo ´ i cˇa . p c´ac d¯ı ˙’ nh biˆe ˙’ u diˆe ˜ n mˆo ´ i quan hˆe . gi˜u . a ch´ung. Nh˜u . ng h`ınh nhu . thˆe ´ thu . `o . ng go . i l`a c´ac d¯ˆo ` thi . . Mu . c d¯´ıch cu ˙’ a gi´ao tr`ınh n`ay cung cˆa ´ p nh˜u . ng kiˆe ´ n th´u . c co . ba ˙’ n d¯ˆe ˙’ nghiˆen c´u . u c´ac d¯ˆo ` thi . . C´ac d¯ˆo ` thi . xuˆa ´ t hiˆe . n trong nhiˆe ` u l˜ınh vu . . c v´o . i c´ac tˆen go . i kh´ac nhau: “cˆa ´ u tr´uc” trong cˆong tr`ınh xˆay du . . ng, “ma . ch” trong d¯iˆe . n tu . ˙’ , “lu . o . . c d¯ˆo ` quan hˆe . ”, “cˆa ´ u tr´uc truyˆe ` n thˆong”, “cˆa ´ u tr´uc tˆo ˙’ ch´u . c” trong x˜a hˆo . i v`a kinh tˆe ´ , “cˆa ´ u tr´uc phˆan tu . ˙’ ” trong ho´a ho . c, vˆan vˆan. Do nh˜u . ng ´u . ng du . ng rˆo . ng r˜ai cu ˙’ a n´o trong nhiˆe ` u l˜ınh vu . . c, c´o rˆa ´ t nhiˆe ` u nghiˆen c´u . u xung quanh l´y thuyˆe ´ t d¯ˆo ` thi . trong nh˜u . ng nˇam gˆa ` n d¯ˆay; mˆo . t nhˆan tˆo ´ chu ˙’ yˆe ´ u g´op phˆa ` n th´uc d¯ˆa ˙’ y su . . ph´at triˆe ˙’ n d¯´o l`a xuˆa ´ t hiˆe . n c´ac m´ay t´ınh l´o . n c´o thˆe ˙’ thu . . c hiˆe . n nhiˆe ` u ph´ep to´an v´o . i tˆo ´ c d¯ˆo . rˆa ´ t nhanh. Viˆe . c biˆe ˙’ u diˆe ˜ n tru . . c tiˆe ´ p v`a chi tiˆe ´ t c´ac hˆe . thˆo ´ ng thu . . c tˆe ´ , chˇa ˙’ ng ha . n c´ac ma . ng truyˆe ` n thˆong, d¯˜a d¯u . a d¯ˆe ´ n nh˜u . ng d¯ˆo ` thi . c´o k´ıch thu . ´o . c l´o . n v`a viˆe . c phˆan t´ıch th`anh cˆong hˆe . thˆo ´ ng phu . thuˆo . c rˆa ´ t nhiˆe ` u v`ao c´ac thuˆa . t to´an “tˆo ´ t” c˜ung nhu . kha ˙’ nˇang cu ˙’ a m´ay t´ınh. Theo d¯´o, gi´ao tr`ınh n`ay s˜e tˆa . p trung v`ao viˆe . c ph´at triˆe ˙’ n v`a tr`ınh b`ay c´ac thuˆa . t to´an d¯ˆe ˙’ phˆan t´ıch c´ac d¯ˆo ` thi . . C´ac phu . o . ng ph´ap phˆan t´ıch v`a thiˆe ´ t kˆe ´ c´ac thuˆa . t to´an trong gi´ao tr`ınh cho ph´ep sinh viˆen c´o thˆe ˙’ viˆe ´ t dˆe ˜ d`ang c´ac chu . o . ng tr`ınh minh ho . a. Gi´ao tr`ınh d¯u . o . . c biˆen soa . n cho c´ac d¯ˆo ´ i tu . o . . ng l`a sinh viˆen To´an-Tin v`a Tin ho . c. Gi´ao tr`ınh su . ˙’ du . ng ngˆon ng˜u . C d¯ˆe ˙’ minh ho . a, tuy nhiˆen c´o thˆe ˙’ dˆe ˜ d`ang chuyˆe ˙’ n d¯ˆo ˙’ i sang c´ac ngˆon ng˜u . kh´ac; v`a do d¯´o, sinh viˆen cˆa ` n c´o mˆo . t sˆo ´ kiˆe ´ n th´u . c vˆe ` ngˆon ng˜u . C. Ngo`ai ra, hˆa ` u hˆe ´ t c´ac chu . o . ng tr`ınh thao t´ac trˆen cˆa ´ u tr´uc d˜u . liˆe . u nhu . danh s´ach liˆen kˆe ´ t, nˆen d¯`oi ho ˙’ i sinh viˆen pha ˙’ i c´o nh˜u . ng k˜y nˇang lˆa . p tr`ınh tˆo ´ t. Gi´ao tr`ınh bao gˆo ` m ba ˙’ y chu . o . ng v`a mˆo . t phˆa ` n phu . lu . c v´o . i nh˜u . ng nˆo . i dung ch´ınh nhu . sau: • Chu . o . ng th´u . nhˆa ´ t tr`ınh b`ay nh˜u . ng kh´ai niˆe . m cˇan ba ˙’ n vˆe ` d¯ˆo ` thi . . • Chu . o . ng 2 tr`ınh b`ay nh˜u . ng sˆo ´ co . ba ˙’ n cu ˙’ a d¯ˆo ` thi . . ´ Y ngh˜ıa thu . . c tiˆe ˜ n cu ˙’ a c´ac sˆo ´ n`ay. 7 • Chu . o . ng 3 t`ım hiˆe ˙’ u b`ai to´an t`ım d¯u . `o . ng d¯i ngˇa ´ n nhˆa ´ t. • Chu . o . ng 4 d¯ˆe ` cˆa . p d¯ˆe ´ n kh´ai niˆe . m vˆe ` cˆay. ´ U . ng du . ng cu ˙’ a cˆay Huffman trong n´en d˜u . liˆe . u. Ngo`ai ra xˆay du . . ng c´ac thuˆa . t to´an t`ım cˆay bao tr`um nho ˙’ nhˆa ´ t. • B`ai to´an Euler v`a b`ai to´an Hamilton v`a nh˜u . ng mo . ˙’ rˆo . ng cu ˙’ a ch´ung s˜e d¯u . o . . c n´oi d¯ˆe ´ n trong Chu . o . ng 5. • Chu . o . ng 6 nghiˆen c´u . u c´ac t´ınh chˆa ´ t phˇa ˙’ ng cu ˙’ a d¯ˆo ` thi . ; v`a cuˆo ´ i c`ung • Chu . o . ng 7 t`ım hiˆe ˙’ u c´ac b`ai to´an trˆen ma . ng vˆa . n ta ˙’ i. Ngo`ai ra, phˆa ` n phu . lu . c tr`ınh b`ay c´ac cˆa ´ u tr´uc d˜u . liˆe . u v`a nh˜u . ng thu ˙’ tu . c cˆa ` n thiˆe ´ t d¯ˆe ˙’ d¯o . n gia ˙’ n ho´a c´ac d¯oa . n chu . o . ng tr`ınh minh ho . a c´ac thuˆa . t to´an d¯u . o . . c tr`ınh b`ay. Gi´ao tr`ınh d¯u . o . . c biˆen soa . n lˆa ` n d¯ˆa ` u tiˆen nˆen khˆong tr´anh kho ˙’ i kh´a nhiˆe ` u thiˆe ´ u s´ot. T´ac gia ˙’ mong c´o nh˜u . ng d¯´ong g´op t`u . ba . n d¯o . c. Tˆoi xin ca ˙’ m o . n nh˜u . ng gi´up d¯˜o . d¯˜a nhˆa . n d¯u . o . . c t`u . nhiˆe ` u ngu . `o . i m`a khˆong thˆe ˙’ liˆe . t kˆe hˆe ´ t, d¯ˇa . c biˆe . t l`a c´ac ba . n sinh viˆen, trong qu´a tr`ınh biˆen soa . n gi´ao tr`ınh n`ay. D - `a La . t, ng`ay 5 th´ang 3 nˇam 2002 PHA . M Tiˆe ´ n So . n 8 Chu . o . ng 1 D - a . i cu . o . ng vˆe ` d¯ˆo ` thi . 1.1 D - i . nh ngh˜ıa v`a c´ac kh´ai niˆe . m 1.1.1 D - ˆo ` thi . c´o hu . ´o . ng D - ˆo ` thi . c´o hu . ´o . ng G = (V, E) gˆo ` m mˆo . t tˆa . p V c´ac phˆa ` n tu . ˙’ go . i l`a d¯ı ˙’ nh (hay n´ut) v`a mˆo . t tˆa . p E c´ac cung sao cho mˆo ˜ i cung e ∈ E tu . o . ng ´u . ng v´o . i mˆo . t cˇa . p c´ac d¯ı ˙’ nh d¯u . o . . c sˇa ´ p th´u . tu . . . Nˆe ´ u c´o d¯´ung mˆo . t cung e tu . o . ng ´u . ng c´ac d¯ı ˙’ nh d¯u . o . . c sˇa ´ p th´u . tu . . (a, b), ta s˜e viˆe ´ t e := (a, b). Ch´ung ta s˜e gia ˙’ su . ˙’ c´ac d¯ı ˙’ nh d¯u . o . . c d¯´anh sˆo ´ l`a v 1 , v 2 , . . . , v n hay gia ˙’ n tiˆe . n, 1, 2, . . . , n, trong d¯´o n = #V l`a sˆo ´ c´ac d¯ı ˙’ nh cu ˙’ a d¯ˆo ` thi . . Nˆe ´ u e l`a mˆo . t cung tu . o . ng ´u . ng cˇa . p c´ac d¯ı ˙’ nh d¯u . o . . c sˇa ´ p th´u . tu . . v i v`a v j th`ı d¯ı ˙’ nh v i go . i l`a gˆo ´ c v`a d¯ı ˙’ nh v j go . i l`a ngo . n; cung e go . i l`a liˆen thuˆo . c hai d¯ı ˙’ nh v i v`a v j . Ch´ung ta s˜e thu . `o . ng k´y hiˆe . u m = #E−sˆo ´ ca . nh cu ˙’ a d¯ˆo ` thi . G. C´ac ca . nh thu . `o . ng d¯u . o . . c d¯´anh sˆo ´ l`a e 1 , e 2 , . . . , e m . Mˆo . t c´ach h`ınh ho . c, c´ac d¯ı ˙’ nh d¯u . o . . c biˆe ˙’ u diˆe ˜ n bo . ˙’ i c´ac d¯iˆe ˙’ m, v`a e = (v i , v j ) d¯u . o . . c biˆe ˙’ u diˆe ˜ n bo . ˙’ i mˆo . t cung nˆo ´ i c´ac d¯iˆe ˙’ m v i v`a v j . Mˆo . t cung c´o gˆo ´ c tr`ung v´o . i ngo . n go . i l`a khuyˆen. Nˆe ´ u c´o nhiˆe ` u ho . n mˆo . t cung v´o . i gˆo ´ c ta . i v i v`a ngo . n ta . i v j th`ı G go . i l`a d¯a d¯ˆo ` thi . v`a c´ac cung tu . o . ng ´u . ng go . i l`a song song. D - o . n d¯ˆo ` thi . c´o hu . ´o . ng l`a d¯ˆo ` thi . khˆong khuyˆen trong d¯´o hai d¯ı ˙’ nh bˆa ´ t k`y v i v`a v j c´o nhiˆe ` u nhˆa ´ t mˆo . t cung (v i , v j ). Chˇa ˙’ ng ha . n, d¯ˆo ` thi . trong H`ınh 1.1 c´o cung e 8 l`a khuyˆen; c´ac cung e 4 v`a e 9 l`a song song do c`ung tu . o . ng ´u . ng cˇa . p d¯ı ˙’ nh v 3 v`a v 4 . 9 .  Giáo trình ĐẠI CƯƠNG ĐỒ THỊ Mu . c lu . c L`o . i n´oi d¯ˆa ` u 7 1 D - a . i