Graf Planar (Planar (

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1 // Grph (Cont) :Apliksi Grph Ssi Grf Plnr (Plnr ( Grph) n Grf Bing (Pln Grph) -ont Rumus Eulr : n + f = imn f = jumlh wilyh = jumlh sisi n = jumlh simpul Ex: Brp jumlh wilyh grf rikut ini? R R R R R R Solusi: = n n = 7, mk f = 7 + = Grf Plnr (Plnr ( Grph) n Grf Bing (Pln Grph) -ont Grf Plnr (Plnr ( Grph) n Grf Bing (Pln Grph) -ont P grf plnr srhn trhuung ngn f wilyh, n uh simpul, n uh sisi (ngn > ) sllu rlku ktiksmn rikut: f/ n Ex: Grf rikut ini lh grf plnr krn ()/ n () R R R R R R Ex: Bgimn ngn grf rikut ini = 9, n = shingg 9 ()() = (nr, n ) phl grf trsut ukn grf plnr! Untuk mngtsi hl ini, ut sumsi ru: stip rh p grf plnr itsi olh pling sikit mpt uh sisi. Dri pnurunn rumus iprolh Shingg 9 ()() = grf plnr. n (slh) yng rrti grf trsut ukn

2 // Lintsn n Sirkuit Eulr Lintsn Eulr ilh lintsn yng mllui msing-msing sisi i lm grf tpt stu kli. Syrt: Grf trhuung Mmiliki u uh simpul rrjt gnjil tu tik simpul yng rrjt gnjil Sirkuit Eulr ilh sirkuit yng mlwti msing-msing sisi tpt stu kli. Syrt: Grf trhuung Smu simpul p grf rrjt gnp Grf yng mmpunyi sirkuit Eulr isut grf Eulr (Eulrin grph). Grf yng mmpunyi lintsn Eulr inmkn jug grf smi-eulr (smi-eulrin grph) Torm Lintsn n Sirkuit Eulr. Grf tik rrh mmiliki lintsn Eulr jik n hny jik trhuung n mmiliki u uh simpul rrjt gnjil tu tik simpul rrjt gnjil sm skli. Grf tik rrh G lh grf Eulr (mmiliki sirkuit Eulr) jik n hny jik stip simpul rrjt gnp. (Not: grf yng mmiliki sirkuit Eulr psti mmpunyi lintsn Eulr, ttpi tik slikny). Grf rrh G mmiliki sirkuit Eulr jik n hny jik G trhuung n stip simpul mmiliki rjt-msuk n rjt-klur sm. G mmiliki lintsn Eulr jik n hny jik G trhuung n stip simpul mmiliki rjt-msuk n rjt-klur sm kuli u simpul, yng prtm mmiliki rjt-klur stu lih sr rjtmsuk, n yng ku mmiliki rjt-msuk stu lih sr ri rjt-klur Stui Ksus Lintsn n Sirkuit Eulr Stui Ksus Lintsn n Sirkuit Eulr Mnkh ri grf-grf tik rrh rikut ini yng mmiliki lintsn n/tu sirkuit ulr. Jik tulis lintsn n/tu sirkuit ulrny Mnkh ri grf-grf rrh rikut ini yng mmiliki lintsn n/tu sirkuit ulr. Jik tulis lintsn n/tu sirkuit ulrny () () () () () (f) 7 f g () () () 7 f

3 // Lintsn n Sirkuit Hmilton Torm Lintsn n Sirkuit Hmilton 9 Lintsn Hmilton ilh lintsn yng mllui tip simpul i lm grf tpt stu kli. Sirkuit Hmilton ilh sirkuit yng mllui tip simpul i lm grf tpt stu kli, kuli simpul sl (skligus simpul khir) yng illui u kli. Grf yng mmiliki sirkuit Hmilton inmkn grf Hmilton, sngkn grf yng hny mmiliki lintsn Hmilton isut grf smi-hmilton. Syrt ukup (ji ukn syrt prlu) supy grf srhn G ngn n ( ) uh simpul lh grf Hmilton ilh il rjt tip simpul pling sikit n/ (yitu, (v) n/ untuk stip simpul v i G). Stip grf lngkp lh grf Hmilton. Di lm grf lngkp G ngn n uh simpul (n ), trpt (n - )!/ uh sirkuit Hmilton.. Di lm grf lngkp G ngn n uh simpul (n n n gnjil), trpt (n - )/ uh sirkuit Hmilton yng sling lps (tik sisi yng ririsn). Jik n gnp n n, mk i lm G trpt (n - )/ uh sirkuit Hmilton yng sling lps Stui Ksus Lintsn n Sirkuit Hmilton Stui Ksus Lintsn n Sirkuit Hmilton Smiln nggot suh klu rtmu tip hri untuk mkn sing p suh mj unr. Mrk mmutuskn uuk smikin shingg stip nggot mmpunyi ttngg uuk r p stip mkn sing. Brp hri pngturn trsut pt ilksnkn? Solusi : rsrkn torm kmpt mk jumlh pngturn tmpt uuk yng r lh (9 - )/ = Brp grf pt mngnung sirkuit Eulr n sirkuit Hmilton skligus, mngnung sirkuit Eulr ttpi tik mngnung sirkuit Hmilton, mngnung sirkuit Eulr n lintsn Hmilton, mngnung lintsn Eulr mupun lintsn Hmilton, tik mngnung lintsn Eulr nmun mngnung sirkuit Hmilton, n sginy. Dri grf-grf rikut ini mnkh yng mmnuhi prnytn trsut

4 // Shortst Pth Prolms W n ssign wights to th gs of grphs, for xmpl to rprsnt th istn twn itis in rilwy ntwork: Toronto 7 Boston Chigo Nw York Suh wight grphs n lso us to mol omputr ntworks with rspons tims or osts s wights. On of th most intrsting qustions tht w n invstigt with suh grphs is: Wht is th shortst pth twn two vrtis in th grph, tht is, th pth with th miniml sum of wights long th wy? This orrspons to th shortst trin onntion or th fstst onntion in omputr ntwork Dijkstr s Algorithm Thorm: Dijkstr s lgorithm fins th lngth of shortst pth twn two vrtis in onnt simpl unirt wight grph. Dijkstr s lgorithm is n itrtiv prour tht fins th shortst pth twn to vrtis n in wight grph. It pros y fining th lngth of th shortst pth from to sussiv vrtis n ing ths vrtis to istinguish st of vrtis S. Th lgorithm trmints on it rhs th vrtx. Thorm: Dijkstr s lgorithm uss O(n ) oprtions (itions n omprisons) to fin th lngth of th shortst pth twn two vrtis in onnt simpl unirt wight grph Dijkstr s Algorithm Exmpl Dijkstr s Algorithm XI - XI - Stp

5 // Exmpl Dijkstr s Algorithm Exmpl Dijkstr s Algorithm () () (, ) (, ) () () (, ) XI - 7 Stp XI - Stp Exmpl Dijkstr s Algorithm Exmpl Dijkstr s Algorithm () (, ) (,, ) ) () (, ) (,, ) ) () (, ) () (,,, ) (, ),, ) XI - 9 Stp XI - Stp

6 // Exmpl Dijkstr s Algorithm Exmpl Dijkstr s Algorithm () (, ) (,, ) ) () (, ) (,, ) ) (, (,,,,,, ) ) (, (,,,,,, ) ) () (, ),, ) () (, ),, ) XI - Stp XI - Stp Stui Ksus Dijkstr s Algorithm () Stui Ksus Dijkstr s Algorithm () Tntukn lintsn trpnk ri simpul k simpul yng lin ri grf rikut ini Solusi :

7 // Stui Ksus Dijkstr s Algorithm () Th Trvling Slsmn Prolm Tntukn lintsn trpnk ri simpul k simpul yng lin ri grf rikut ini Sn Frnsiso () Los Angls () Dnvr() 7 Chigo() Nw Orlns() 9 Mimi(7) Boston() Nw York() XI - Th trvling slsmn prolm is on of th lssil prolms in omputr sin. A trvling slsmn wnts to visit numr of itis n thn rturn to his strting point. Of ours h wnts to sv tim n nrgy, so h wnts to trmin th shortst pth for his trip. W n rprsnt th itis n th istns twn thm y wight, omplt, unirt grph. Th prolm thn is to fin th iruit of minimum totl wight tht visits h vrtx xtly on mnntukn sirkuit Hmilton yng mmiliki oot minimum Ex: Trvling Slsmn Prolm Th Trvling Slsmn Prolm Exmpl:Wht pth woul th trvling slsmn tk to visit th following itis? Toronto Chigo 7 7 Nw York Boston 7 XI - 7

8 // 9 Th Chins Postmn Prolm Dikmukkn olh Mi Gn (rsl ri Cin) p thun 9. Mslhny lh sgi rikut: sorng tukng pos kn mngntr surt k lmt-lmt spnjng jln i sutu rh. Bgimn i mrnnkn rut prjlnnny supy i mlwti stip jln tpt skli n kmli lgi k tmpt wl krngktn. Solusi mnntukn sirkuit Eulr i lm grf Ex: A F B C E Lintsn yng illui tukng pos: A, B, C, D, E, F, C, E, B, F, A D Rfrnsi. Rinli Munir, Mtri Kulih Mtmtik Diskrit,Informtik- ITB, Bnung,. Rinli Munir, Mtmtik Diskrit,Informtik, Bnung,