Ve) Gueaphx | > A Paph G= (VE) twonists of 8 sek of VerHaus vig Edepes le) here VY finite non enepty Bet of verte (v9 ES in se ‘bdgese) , ” Ge yey Eq t- Nad £ Ns yey *y tw Vv, * Vo, q aw mE connected b v ei Voy, lL L Cm f yap, Yak Vy" IM, Yo, Vey M4) 3 vertices. [Ms v09, War var, Ova, ¥4 9, CVA MD Ce Mg) C209] bo ee 6 a Vi — Va 4, Ye “ . 23 M1, Vs © qi yo ve YS les 7 toy ** 3 v v +a a? as VG M2 YeDey ak RUT Teme A: nia bs {Oe at] EBB HB Nodss > vertices Eagts > Lundy bho 2 vethees (010 2 nodes, 2) yee?) Graphs jhe arc Types 5} Birapl. Shay are! Q) Durected 9) cycle by unduurctid n) DAG °) wargnicd HeRR Wy) puget rf ay Computed 4d) gol} oop 2) subgeaph k) Bi-eonnecti ty L) eut verux | = path | ly Auiccted graph | > Aireted edge SMV (eM) | (vy, ved4 Ca wd | Cvs, va) Fg, 2) | (v2, OY 2%)b) Unde cled uaph Undireoted edges rages ee anced b woe fwd (YW) = lv, wy (v,, vo) = Cv.) Ova v5) = (vg, va) , x G wha e | Ya, W) =) CW, vp y |e) ) waltg tid quark (vate ne aa [2a waged FP i: guagh woliick, + em os song ln A edi | Mt eiiig9 49 ‘ang od a valuea) completa graph me seed 7 ait an undiieeted qpeph ohn veabiees consist % 1D) nin +N, of edges, a? Strongly wnnsded qorh > 24: Socal niw. 5 Ce file ahding 2). sub-qaphs Carpet) DA Subgraph “Boh aph Gu a Graph Auch that thu Aak Of yeti & sok of edges of 6 ' are proper subse of the et of elas ef 4, @) j —_¥ Q! f) path = (amour) | DA path is aenottd witha. sequence, of Wei ether ok an edges me vertem at yeatex.6) @) ‘ Wi, Meu! areabing } vy » “e E> no. of | )—-8 et & ps 9) cycle \ (ret | boop) Jy A tlosed talk thootigh the raph with | . , ; ‘yepeated vertion mastty hanng the tame start \P » eaetty hawng Foe verter % ending yetesr , | ) >) Vv) Ove 2% Vy. Vo Vy OV, OV Vy Vp V2 ON Vyas 03 Vg 25 OY v hy V3.2 3 7 V5" sp acyte yeph: = @ > edges ane qeere ahould | fomt a at eyele 5 WEA + 2B Adtected Aeyelic graph > No wyele. iaetins Auyelte graph DTt doen't fom a cycle. ‘ ted hp we eonneeted 1) strongly tanneeted Gop alte L% U A ~ path > AW nol” peachadk > Aimed geek) Aeger we Verte dequs (wv) 2 types oh degree! a) Indegve b) outdenres a) Videgree ~> vedio! the NO- Of edges Thay , ungide Weldentty That ventex ¥) outdegue -) yeuten i the tobe north edac, ne Una, faa man Rey [eri] andi ey May vo | My Na Ma | NS 4 ie Va Va ae ve | va | Mi a |¥) conneetid qark: 2 an undirected qiaph , ws said ¢ be connected i for we par of distant vation vj #Y un WEG) tere wo qsaph fiom vj b wy inh aL x £) Comporunt > the martmal connected sespoph th n) Biconnectot > preenmeled Y cannot be broken ant connected ingle age oph are the qeaphs ‘[ohich 2 dfs connected pices bgr secusecesessesess © “4 ye, 3 Fs E —& OE | ©) cut vertex: ‘ | 13 A vertex V ws a undliechd graph & 4 fp | A ‘ he | [emery Uk aliseonneets the gap aut yetea & also called ast elation | | @. @.1'| 6 ~Os¢ | ie eae @ | 6 Representation h Graph: : . there one > Upes 8} Crvaphs. They ane: point 5 3. | yy Adjacent Matrix (AM) | ) Adjacent nat (Ab) i) Am: . ; | DA graph tontatning n vetie can be Seo by a matia with n apws¢ n columns. Matra formed by sown the eres tocegntad | ats a wows 4 J™ eolumns oh tre matin between_veattea * the qrerh i... Frample 1. Given the Ama AL ‘Of he tip Letui qiaph: . , 9 et re th : ALEl Cl PEE! | verte, ol] tli|olo (mae: 5, FD | Bli jo |nott a4 Arad wpba | c =D |ne | @. I ads | Dlofojejo|r|! ot le | of ofolo| oly te le | of fo|o| ofo rane ie 6 "BE ELE a [mle io fe [> | wiv =>. [Raketed graph onal ded quaphl| wilgnerd graph] \aitectein - ie \ ) | | oe lure . ae | (ver) che \ : | (Mir Vs) My Ya, N3 io (2,0) Ya 4 ve 2 (210d . | tA % vp | (v1, ¥2) | | ‘4 V3 3 {iy | ] (24%) ‘ 5 —__ ¥% 2U3rn) Ve, tay (¥,,¥3) (va }va) \W21 Va) (vy v3) (yy, ¥2) (r,¥, (v5, va) 7D (4, v3? —‘ piare 4 bl AL: ian he DA graph See m veties 4 n edaec can be. repursanted uring? a a uiked Usk oeGeary Traversac (AT) ST k a graph acehnd que sed ‘for a verter th agraph. mY RAN ’ | > Te is used to aeelde te ander Pf verbiees uy iceted un the seach prec. There ane >-ty pes of Search din GT. They are: ") Breadth first Starch (BES) |) top ss tars) \ |. Breadth fiast Search (BFS): | °>) aby i “o rprretes °*b pare each nooles oF the _qeaph unto 2 parks. vated Nob - vested > The purpose fh che algon thm wes we eit Alt Hha_wustn_estile evel oyetes. DIE usu A ple OS bo Aepreent me yesttid necles, DT works duel sy eel > dnitiatip, BFS stab ak a qrco rectexechuth Js ak evel_ 0, air put hr at wads att the verhices al level t terel 0,19, 3% 418, 6, Fe- ete: urls o & Som. 8 A HSE yee tepad — > Source - destination > Find Me Ahoatut path - a) fiskstras Apert edi method & tend he eo sete, 2B Are shortest path problem. | Buk kuwownh abgodttin “to Jad ee atak pati ain quark: ' 25 Appureable to both quaphs with wore agate weight nlp. > aingte - sowie shortest path : © for a dined + tuncldreeted un verte® called the Source th . wuignted connected graphs ond Svortak path Jy aut ie other verti es:steps ‘ to renter neortsl ty le Fund, the Sp fii doer ue, noob 9. ‘Then, eptind tte SP dum dou bo (he Ch js we Neartert yeater. ‘a © procs for tel eth ov vertices gy. Contfrue nares? to the Aotvrte. MAE rapt ag iy | >, Forws a subir wrty These, Veatt ed, Sie. yes ealges of the shorted path . Example : ‘ (D Apk-. Bea, ty, Cas) alaalauraieh es Fl -y00) Sop oe A LAY (ee 7 a> be > 9.|/b(a,1) | Clb, 399 penta 00302 babe as bod Cb, 8) > bE ACO F272, ' ue ; \ ecb bie OCU) FEE wen Toph bed Els #(-,a) —+— Ly SP? eC b,29 3) etb, 2) "| €(b, a) — a fle, 4) watieapts #(e4) gp ab, 2) ath,2) | a4) csp Hd)eee 6 no vertea 1 Hay ay | val es ru ee ae S gbutye 8 21! o7 =3 aba» abbrd= 3 alte ee => a pee # oe) l41eh 2 & oc i = L ~ ” ~ atta &@ c.3.p frm Aowtse can be qin! but by kag bstias algosthm © pid the s.p bebwen node Ate bt wr He tellowing Figs ty Applydg’ Ary ketal agent, 1 poradjacent noc - 9, date tost ds tompute por | (A) 3N pr “ey : wp Rete dst 2 3.P DRDO DEDH sp ah 9B et 3H Spa 47b>t-5H ee ” ~3 MINION SPANNING “TREES. 2 nny Ree oh a qeph & a subgraph Wak corrtaans AU the vertices 2 uk alo a Wee +: at qeph ee have mawp pening Ne hee . taeasn?t frome tgete). y fon tude stupa gragh we ) eas Bray mucitiple speeeny bites . \ ‘r be b8. ie Conagtien : |b To find mininawm Span tee ds an tna ted graph, we assume that he Ten graph | ond weighted paph- Ea mst % an undiuedtd graph & ts a hee jormed “farm agh ne that convert All the verticw ‘ 4 with minimum total cost -Q eae % ie wiilate! Homnun 1 prims | ovttiim | ere een 2. Kruskal 4 |. pam’ Ayrithen > P, cs nthm . aaa a art quer alge PTE EB Used for Fndidg the met [rinime ie a Te Tree] +} a geen geaph. 7 fo “pry PA | the given quaph retest be coskghted, tonmectid 2 unctitedicl. Ok tn ts Aimplemenita tion » 1, Randomly choose ony Vertex | 2. The vertex Comnectitig t the oly hating! hast” twiighted is cauatly selected), 3. Pad all the edges that correct the tree bp WLU vertices. AM 4. Fd ‘the deat! ae oage meneant Chtes cages 4 ‘Uneludld ae “4 ges ost vba 5. bh iia) the tae cates avi the dil thak tdi 3s Lek eae : yr 6. keep repeating these slaps inti ald thes yertives are ‘included 4 Met ds obtained for me Neat “east eoetigit -b(a,3) C(-) 00) d(~«) e (a6) Fla,s) sp —» b(a,3) [Need t Ace shortest Youu J Cb, 1) Ato) e(a,6) ost | £0, 92 an 2 oe Sp Cbs!) ped Z ve be ee| ~ set] Thee Ronaliuing yetious vertices q | —+— Selby jafe.ey | (a,b) | Flora) ; | Spo fear) 4) jo tsea ats (4,29 é “MSS SS ep ay ONY: BM | | Ey ) | 5) ay | ates) 628 | SP = d(f,5) 3 + A Ss © | udicosb| os hg + Cost o Cce MST = add all the weigh Of the vertey s” 2 B+ 44 pegeist Bo inekels Aigorith ms ¢~ (kA) = KA ds used t fend Ke mininurm re Ke for % Connected tong beted qaph . 7 0 i ; = The main target ee perm is be Hog the Subseb e ou es by y wg whith we Can trovene ee 4 vertex 4 wf gaph. D It +ollows rey’ a prrach. bps: 1. FySb, dort alt the edges -fyom lov wget Mh Now, take the o edge with Lowest wougnt + add i Ho the 3 tue Fo is mga seat qe te a in ' tie y \ Continue to add the edges untel we reach all vectored ip ws vcated - Example: agp . The weight ¢h the edge ie abore. graph isEPs | ous gee ebeloww | PA ty Le | a8] he qo [ae BC eh wegne |] 4 ro | By 3 ye oO Now) xort “ie cage gr sbeldov wo she eae Ader the weeps nes qe De‘ with waght a to the MST. ck Gy wok acto Tie cycle,[ps a Add , the “age Be ne (8, to Ghe M87, wb ; ie fs (ae ty the var a Now, pick He sigs, amie Y notidvrering eH aks sm eats « chps: hte MOT, te eden « Nows saab db. oat Goats mal tyele » 80 ‘qs o dn dae Lyle ‘snout not be for. 5ae Fe pick the tala AC) with vinight oe Neasit. e. hy nate BL "eyaley 40 listed at [eye ahouta not be fore) | 1 den ; a + Pid ow edge Ab uit 0H * Tnebuding me will aio cate the tydls,"60 Asad 185, heal op 5 Pe obtained avr the Tren wagad graph Oy aig Rata ge E ie! ee Cast op the MST > Edge tonne AB+ Be + Bc pap ” . e oly » Cost of msrp _ o ==