Graph Theory Homework Summer 2018

Transcription

1 Grph Thor Homwork Smmr 20 Bs on Gross n Ylln 2th Eition Jl 2, 20 Contnts Homwork Homwork 02 Homwork 0 Homwork 0 Homwork 0 Homwork 0 Homwork 0 Homwork 0 9 Homwork 09 0 Homwork 0 9 Homwork 0 2 Homwork 2 Tst 2 Homwork 2 Homwork Homwork Homwork Homwork 9 Projt 20 Homwork 2 Homwork 9

2 22 Homwork 20 2 Homwork Homwork Tst 2 20 Ths r prolms will oth il n t th n o lsss. This PDF il ws rt on Jl 2, 20. Homwork 0 Stion 2. Iso s 2.., 2.., 2.. n 2.. n ol qiz rom Smmr 0: Fin ll possil isomorphisms tps o th gin kin o simpl grph: A -rtx tr. 2.. Fin ll possil isomorphisms tps o th gin kin o simpl igrph: A simpl -rtx igrph with xtl or rs. 2.. Fin rtx-ijtion tht spiis n isomorphism twn th two grphs: s z q x w r t 2.. Fin rtx-ijtion tht spiis n isomorphism twn th two igrphs: s z x w r t olq. List ll possil isomorphism tps o trs with gs rll so tht no two trs in or list r isomorphi. olq.2 For th pir o grphs low, i i th r isomorphi or not. I th r isomorphi, thn gi n isomorphism. I th r not isomorphi, thn xplin wh th r not isomorphi. M S T R N X Q P 0 W V U 2

3 2 Homwork 02 Stion 2. Iso s: 2.., 2.., 2.., Explin wh th grphs r not isomorphi. 2.. Isomorphi? pro isomorphism. Non-isomorphi? Explin Isomorphi? pro isomorphism. Non-isomorphi? xplin 2 g i 9 0 h j 2..2 Isomorphi? pro isomorphism. Non-isomorphi? xplin wh not. 2

4 Homwork 0 Stion 2. Mtrix: 2..2, 2.., 2.., Fin th jn mtrix, A G, th inint mtrix I G n th tl I V,E (G) For G low: j g h i w x 2.. Drw th grph with th jn mtrix A. Th rtis r in th orr,,, 0 A = Drw th igrph with th jn mtrix A. Th rtis r in th orr,,, 0 A = For th grph with th jn mtrix A. (Th rtis r in th orr,,, ) 2 A = Drw th grph G n ompt A 2 n show A 2 [, ] is th nmr o pths o lngth two in G rom to itsl. Show A 2 [, ] os th sm or pths o lngth two rom to. Homwork 0 Stions. n. Dgrs:.. (hint pg 22),..,..2,.... Formlt th prsonnl-ssignmnt prolm [Applition..] s mximm low prolm (Hint: n rtiiil sor n sink to th iprtit grph) (Hint pg 22).. A 20-rtx grph hs 2-gs. Er rtx hs gr or. How mn rtis h gr?..2 Eithr rw -rglr -rtix grph or pro tht non xits... Pro tht no -rtx -g simpl grph hs imtr grtr thn 2.

5 Homwork 0 Stion.- Trs :..,.2.2,.2.,..2.. Eithr rw th sir grph or xplin wh no sh grph xists: A 9 rtx, 2-omponnt, simpl grph with xtl 0 gs n 2 ls..2.2 Drw ll root tr tps with rtis. How mn irnt grph isomorphism tps o th rprsnt?.2. Drw ll possil inr trs o hight whos intrnl rtis h xtl 2 hilrn. Grop ths into lsss o isomorphi root trs...2 Gi ll-orr, prorr, inorr n postorr trnrsls o th inr tr low: Homwork 0 Stion.- Trs 2:..,..0,..,..2.. Rprsnt th gin rithmti xprssion n xprssion tr. Thn gi th prix n postix nottions (( ( + ) (g ))/( + )) ((g + ) h)..0 Gi n xmpl o two -rtx inr trs whos ll-orr n pr-orr r,,, t whos post-orr is not... Using th srh tr low, o th oprtion thn rw th srh tr. Insrt 0, lt 20, insrt How mn irnt insrtion sqns o th ks,, 9, 9,, 9 r thr tht rslt in ln srh tr?

6 Homwork 0 Stion.- Trs :..2,..,..,..2 n ol qiz 2 rom Smmr 0: Do sing g 0.. Constrt Hmn tr or th gin list o smols n wights. Cllt its rg wight lngth n no th wors: n ggg. Us lt-to-right orring to rk tis. lttr g h rqn Insrt 0, insrt, lt 2 into th priorit tr Drw th tr tr r oprtion...2 Drw th priorit tr or th gin sqn or xplin wh it is not priorit tr: 0, 20, 2,,, 2,,,,,.

7 olq2. For th xprssion tr + / 9 9. List th rtis in ll orr 2. List th rtis in prorr. List th rtis in inrorr. List th rtis in postorr. Compt n simpli th omplx nmr gin th tr. olq2.2 Th omplt inr tr o hight 2 hs rtis.. How mn insrtions sqns (prmttions) o, 2,,,,, whn insrt into n initill mpt Priorit tr rslt in omplt inr tr? 2. How mn insrtion sqns (prmtions) o, 2,,,,, whn insrt into n initill mpt BST (Binr Srh Tr) rslt in omplt inr tr? Homwork 0 Stion. n.9 Trs...2,..,.9., Eno th ll tr low s Prür sqn 2.. Constrt th ll tr with Prür sqn {2,,,,, }.9. Drw r tr T sh tht th g omplmnt T is tr..9.2 Wht r th minimm n mximm inpnn nmrs o n n-rtx tr?

8 9 Homwork 09 Spnning Trs:.2.,.2.,..,.. n th onlin ssignmnts rom (with sr or si n psswor or mpi:) Strt rom rtx, o pth-irst srh sing Algorithm.2. inling th s nmrs. First s lxiogrphil orr s th lt priorit, thn rpt sing rrs lxiogrphil orr. Us th grph low. w o x m r i s l j n z p k g t h.2. Strt rom rtx, o rth-irst srh sing Algorithm.2.2 inling th isor nmrs. First s lxiogrphil orr s th lt priorit, thn rpt sing rrs lxiogrphil orr. Us th grph o... Appl Prim s Algorithm to th wight grph low to strting with th rtx s n rsoling tis lik in Exmpl.., lxiogrphi orr irst non-tr rtx, thn tr rtx. Drw th rslting tr n gi th totl wight. s g 9 0 h.. Appl Dijkstr s Algorithm to th wight grph o to strting with th rtx s n rsoling tis lik in Exmpl.., lxiogrphi orr irst non-tr rtx, thn tr rtx. Drw th rslting tr n gi th totl wight.

9 0 Homwork 0 DFS pplitions:..,.., Fin th loks n t rtis o th grph in..,..9.. Gi th DFS nmr n th low nmr or h rtx or pth-irst srh o th grph low, strting rom rtx. Us lphtil orr s th lt priorit. Vri non-root rtx is t-rtx, i n onl i, hs hil w so tht low(w) snmr(). h g i.. Th sm grph o strting rom rtx Bloks n Ct rtis Fin th loks n t-rtis o th grph in [..]..9 A onnt simpl grph G with mor thn 2 rtis with mimimm gr δ thn G ontins l o lngth grtr thn δ. Show this is not tr or non-simpl grphs. 9

10 Homwork Elrin Trls:..,..,..2,.. n th onlin ssignmnts with sr o si n psswor or mpi: Whih whls W n r Elrin... Appl Algorithm,.. to in n Elrin tor in th grph low 0 m r q t l g h p i j n k..2 Us th moii Algorihm.. rom Exris..0 to ontrt n opn Elrin tril in grph low. g h i j m k.. Us n pproprit moiition o Algorithm.. to in n Elrin tor in th igrph low i m j g h k 0

11 2 Homwork 2 Stion.2 Elrin Applitions:.2.2,.2,,.2,2,.2.2 n prolms n on tst rom Smmr 0 Drw th (2, )-Brijn igrph n s it to onstrt two irnt (2, )-Brijn sqns..2. Whih rtis o th Brijn igrphs h sl-loops. Jsti or nswr..2.2 Us Algorithm.2.2 to in minimm-wight postmn tor or th wight grph low Us th mthos o Applition.2. to in n RNA hin whos G n UC rgmnts r s gin G rgmnts: CCG, G, UCCG, AAAG. UC rgmnts: GGAAAG, GU, C, C, C, C. tst. How mn gs? os th simpl grph G h i. Compt th nmr. () G hs 0 rtis n ll o gr. () G is li, hs omponnts n 0 rtis () G is omplt inr tr with hight. 2. Compt th nmr n rw grph, i G hs rtis n r rtx hs gr or (Gi ll possiilitis.) tst. Tr or Fls.. Er wlk is tril. 2. I G is onnt n E = V, thn it hs l g so tht G is tr.. An li simpl grph hs E V onnt omponnts.. For n, K n is Elrin n is o.. Er onnt simpl grph with t g hs t rtx.. A simpl grph G with 0 gs n 20 rtis n h (G).. I n 2 n m 2, th grph K n,m hs m + n rtis, mn gs, imtr 2 n no t gs.. A onnt grph with gr sqn ( ) is tr. 9. For h h thr is inr tr o hight h so tht or h non-l rtx, th hight o th lt str o is on pls th hight o th right str o. (Cont mpt strs s hing hight.) 0. Thr r isomorphism tps o 2-r -rtx simpl (loop r) igrphs.

12 Tst No homwork. Homwork Cl Sp:..2..2,.., Consir th tr T with gs {,, g, } in th grph G low. Fin th nmntl sstm o ls ssoit with T. Fin th nmntl sstm o gs-t ssoit with T. w g x z..2 Fin th non-nll lmnts o th l sp W C (G) or th grph G low. Fin th non-nll lmnts o th g-t sp W S (G). x z g w.. Rpt or th grph low: x z w.. Show th olltion {{,, }, {,,, g}, {,, h}} o g ssts o E G orms sis or th l sp W C (G) or th grph G low. Fin irnt sis hoosing som spnning tr n sing th ssoit nmntl sstm o ls. h g 2

13 Homwork Conntiit:..,..,..,..... Eithr rw th grph or show no sh grph xits: A onnt grph with rtis n 0 gs n no t-rtis.. Eithr rw th grph or show no sh grph xits: A -onnt grph with xtl on rig... Dtrmin th rtx n g onntiit o th grph low.. Gi n xmpl o grph G with Homwork Dlit:..2,..,..2,... κ ν (G) < κ (G) = δ(g)..2 Fin th nmr o intrnll ijoint -pths n s th Crtiit o OPtimlit to jsti or nswr in th grph low: t w x s z.. Fin th nmr o intrnll ijoint -pths n s th Crtiit o OPtimlit to jsti or nswr in th grph o:..2 Inti th loks n rw th lok grph or th grph low: t w x s z.. Fin two non-isomorphi onnt grphs with six rtiw, six gs n thr loks.

14 Homwork HAM:..,..[,] (oth n onts s on prolm)..[,] (two s on) Gr o o orr... Whih n-rtx whl grphs, W n r Hmiltonin.. n Drw th grph or pro non xists: An -rtx simpl grph with mor thn gs tht is oth Elrin n Hmiltonin. An -rtx simpl grph with mor thn gs tht is Elrin t not Hmiltonin... n Eithr onstrt Hmiltionin l or pro non xists. g x w.. g i h.. Gr o o orr Constrt th Gr o o orr.

15 Homwork TSP: Appl Algorihtms..2 n.. to th grph o prolms.. n..... Appl Algorithm..2 to th grph low with ost mtrix A A = Appl Algorithm..2 to th grph o with th ost mtrix A = to.. Appl Algorithm.. to th grph o..... to.. Appl Algorithm.. to th grph o... 9 Projt No xtr homwork.

16 20 Homwork Forin Grphs, Elr n Dlit:..,..,..9-o..-o... Fin Krtowski sgrph in.. Fin Krtowski sgrph in..9-o Show V E + F = 2 hols or oth grphs low..-o Eithr rw pln grph tht mts th sription or show non xists A iprtit simpl grph with rtis n gs. A simpl plnr grph with rtis n gs.

17 2 Homwork 9 Strt o igrphs. Critil pth: 2..2, 2.., Appl CPM to trmin th rlist strting tim o h tsk, th rlist ompltion tim o th ntir projt n th ritil tsks. Tsk g h Drtion 0 Prssors - -,,,g 2.. A stnt mst omplt 0 orss low or h or sh n grt in ppli mthmtis. Th orss n thir prrqisits r list in th ollowing tl. Appl CPM to trmin th minimm nmr o smstrs n to grt. Cors C = Clls C2 = Clls 2 DM = Disrt Mth C = Clls A = Algorithms GT = Grph Thor DE = Dirntil Eqtions S = Sttistis P = Proilit LA = Linr Algr Prrqisits Non C C C2 C,DM DM C2 C C2,DM DM,C 2.. Drw ll th isomorphisms tps o simpl igrphs with rtis n rs.

18 22 Homwork 20 Ntworks :..,..,..0,.,2.. Vi tril n rror n th Crtiit o Optimlit to in mximl low n minimm t. w 0 z t s 0 x.. Fin ll s-t ts n inti ll minimm ons. x s z 2 t..0 Trnsorm into singl sor n sink ntwork n sol. Us th Crtiit o Optimlit to onirm. 9 x s s z 2 w t t 2..2 Fin mximl low n miniml t in th ntwork low tr trnsorming th ntwork to on whos pitis r intgrs. Us th Crtiit o Optimlit to onirm. /2 w 2/ 2 z /2 /2 t s / / 2/ /2 x / /2

19 2 Homwork 2 Ntworks 2:.2.2,.2.,.2.,.2. (s ntwork low prolm).2.2 Us th Mximm Flow Algorithm to in mximm low n miniml t in th ntwork low: s 2 2 x z w 2 t.2. Us th Mximm Flow Algorithm to in mximm low n miniml t in th ntwork low: s 20 r x z 2 9 w.2. A ompn mintins thr wrhoss X, Y n Z, n thr stors A, B n C. Th wrhoss h rsptil 00, 00, n 900 mowrs in stok. Thr is n immit mn rom th stors or 00, 00 n 00 mowrs rptil.in th grph mol low, th r pitis rprsnt ppr ons on th nmr o mowrs tht n shipp in singl on tht trnk rot sgmnt. Th immit nos m rgr s trnsshipmnt points, whr trks r hk, rl, mintin t. Cn ll th mn mt? I not, how los n th ompn om to stising th mn? Z w A Y X z B C.2. Th Johnson, Pt, Shrg-Srs n Wr milis r going to th Wintr Prk Siwlk Art Fstil. For rs r ill to trnsport th milis to th show. Th rs n rr th ollowing nmrs o popl: r, or; r 2, thr; r thr; n r, or. Thr r or popl in h mil, n no r n rr mor thn two popl rom n mil. Formlt th prolm o trnsporting th mximm possil nmr o popl to th stil s mximm low prolm. 20 t 9

20 2 Homwork 22 Ntworks :..2,..0,..,.. n th tr ls prolms rom Smmr 09: Fin th lol g-onntiit twn th pir o soli rtis low ining th mxmimm low in n pproprit ntwork...0 Fin th lol rtx-onntiit twn th pir o soli rtis o ining th mxmimm low in n pproprit ntwork... Fin mximim mthing n minimm rtx or or Sppos tht thr r n workrs n n jos to prorm. Eh workr is qlii to prorm xtl k jos, k, n h jo n ssign xtl k workrs. Pro tht h jo n ssign to irnt workr who is qlii or tht jo. TFSm09 Tr or Fls n short rson. Th whl grph W n is sl l. 2. Th ohron hs 2 s, 0 gs n 20 rtis.. Th Ptrsn grph ontins sgrph homomorphi to K.. I th onnt grph G hs E = V + thn th l sp W C (G) hs non-nll tors.. A ntwork with niq mximm low, hs niq minimm t.. I n, m 2 n n + m thn K n,m is non-plnr.. For ll omplt iprtit grphs κ (K n,m ) = δ min (K n,m ). Eh mximl mthing is mximm mthing. 9. For simpl grph G, th minimm nmr o rtis in rtx or o G n stritl iggr thn th mximm nmr o gs in mthing o G. 0. Thr r isomorphism tps o loop-r simpl igrphs with rtis n rs. 2 Tst 2 No homwork. 20