Topic review Topic 9: Undirected graphs and networks

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Gervase Fields
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Topi rviw Topi 9: Undirtd graphs and ntworks Multipl hoi Qustions 1 and 2 rfr to th ntwork shown. 1. Th sum of th dgrs of all th vrtis in th ntwork is: A 5 B 10 C 12 D 13 E 14 2.

1 Topi rviw Topi 9: Undirtd graphs and ntworks Multipl hoi Qustions 1 and 2 rfr to th ntwork shown. 1. Th sum of th dgrs of all th vrtis in th ntwork is: A 5 B 10 C 12 D 13 E Th list of dgs an writtn as: A {A, B, C, D, E} B {(A, A), (B, B), (C, C), (D, D), (E, E)} C {(A, B), (A, C), (A, D), (B, C), (B, D), (C, D), (D, E)} D {(A, B), (A, C), (A, D), (B, C), (C, D), (D, E)} E {(A, B, C, D), (B, A, C, D), (C, A, B, D), (D, A, B, C, E), (E, D)} 3. From th following matrix rprsntation of a ntwork with vrtis A, B, C, D, E, th dgr of vrtx A is: A B C D E Aé ù B C D E ë û A 1 B 2 C 5 D 6 E Using th matrix in qustion 3, th isolatd vrtx is: A A B B C C D D E E 5. A rtain planar graph with 6 vrtis an dividd into 4 rgions (or fas). How many dgs dos it hav? A 2 B 6 C 8 D 10 E Cannot dtrmind from th givn information

6. Considr th ntworks i, ii, iii and iv. Whih ar onntd graphs? i ii iii iv A i only B i, ii and iii C i and iii D ii and iv E All ar onntd. 7.

A C G A H G F E D B A C B E B C G H A B D E B C C C G H A B D E F D A B C D E F G H E Thr is no Hamiltonian path for this graph. 8.

is an vn numr of vrtis D must not start on a vrtx with odd dgr E rquirs th starting vrtx and th nding vrtx to th sam 9.

2 6. Considr th ntworks i, ii, iii and iv. Whih ar onntd graphs? i ii iii iv A i only B i, ii and iii C i and iii D ii and iv E All ar onntd. 7. For th graph shown, whih is a Hamiltonian path? A C G A H G F E D B A C B E B C G H A B D E B C C C G H A B D E F D A B C D E F G H E Thr is no Hamiltonian path for this graph. 8. For th graph in qustion 7, an Eulr trail: A dos not xist aus thr ar 2 vrtis whos dgr is 3 B xists aus thr ar xatly 2 vrtis whos dgr is 3 C xists aus thr is an vn numr of vrtis D must not start on a vrtx with odd dgr E rquirs th starting vrtx and th nding vrtx to th sam 9. Th shortst path from H to E is: A 24 B 25 C 26 D 27 E Th lngth of th minimum spanning tr for th graph in qustion 9 is: A 39 B 40 C 43 D 49 E dpndnt upon th starting vrtx 11. A onntd planar graph has 12 dgs. This graph ould hav: A 5 vrtis and 6 fas B 5 vrtis and 8 fas C 6 vrtis and 8 fas D 6 vrtis and 9 fas E 7 vrtis and 9 fas

3 Short answr 1. For th ntwork in Multipl hoi qustion 9: a lal th vrtis and dgs in trms of th vrtis writ a matrix rprsntation of th ntwork. 2. Draw a diagram of a ntwork rprsntd y V = {A, B, C, D, E, F} and E = {(A, B), (A, C), (B, C), (B, D), (B, E), (C, F), (D, E), (E, F)}. Calulat th sum of th dgrs of all th vrtis. 3. a Convrt th ntwork in th figur to a planar graph. Confirm Eulr s formula for th rsult. 4. Starting at vrtx A, dtrmin an Eulr iruit in th figur shown. 5. Starting at vrtx A, dtrmin two diffrnt Hamiltonian Cyl for th graph in qustion In a town thr ar fiv frinds: Paul, Bn, Kvin, Matt and David. Th frinds houss ar linkd y th numr of footpaths as givn in th matrix. é ù ë û Du to an ovrsight, th nams of all th frinds wr not listd with th rows and olumns; howvr, th following information is known: Bn and Paul hav two footpaths twn thir houss. Thr is only on path twn Kvin and ah of his frinds. Paul and David hav on path linking thir houss to ah othr. Th sond olumn in th matrix rprsntation aov rprsnts Paul s dgs. a Rdraw th matrix with th nams in thir orrt plas aov th olumns and sid th rows. Using th answr to part a, draw th ntwork of footpaths and houss. 7. Us th information in qustion 6 to omplt th following. a Rdraw th ntwork and add a path to it that will nal an Eulr iruit to travlld y th young mn. Why would th frinds wish thr to th xtra footpath in thir ntwork of footpaths? Using th nw ntwork diagram, stat an Eulr iruit.

8. Dtrmin a Hamiltonian path for th ntwork drawn. 9. Th figur rprsnts rail onntions twn 11 towns. A tlgraph systm is to st up onnting all th towns. Find th shortst total distan for this systm.

Thr ar 6 faturs (Oak tr, Kiosk, Flowr ds, Pin tr, Gazo and Lak) and 4 ntrans. Paths onnting various faturs ar drawn as rd lins and an takn as straight lins onnting th dots for ah fatur.

This is not possil, ut, y adding a nw path twn two faturs, it will om possil. Add this nw path and dtrmin th visitor s path. What kind of path is this?

4 8. Dtrmin a Hamiltonian path for th ntwork drawn. 9. Th figur rprsnts rail onntions twn 11 towns. A tlgraph systm is to st up onnting all th towns. Find th shortst total distan for this systm. Not that mssags an rlayd from on station to anothr via a third station. Extndd rspons 1. Th plan rprsnts a otanial gardn in th town of Lovly Banks. Thr ar 6 faturs (Oak tr, Kiosk, Flowr ds, Pin tr, Gazo and Lak) and 4 ntrans. Paths onnting various faturs ar drawn as rd lins and an takn as straight lins onnting th dots for ah fatur. a d Draw this ntwork of 10 vrtis as a planar graph. A visitor arriving at Entran 2 wishs to travl along all paths xatly on. This is not possil, ut, y adding a nw path twn two faturs, it will om possil. Add this nw path and dtrmin th visitor s path. What kind of path is this? Masur th distan along th paths using a rulr and th sal at th ottom of th plan. Quot distans to th narst 5 mtrs. (Do not inlud th path from part.) Find th shortst path that onnts all th vrtis (inluding ntrans). What kind of path is this? Mak a drawing of this path. (Do not inlud th path from part.) Find th shortst path twn Entran 4 and Entran 2. (Do not inlud th path from part.)

2. In a stat forst thr ar six amp sits: Angl Vally, Booming Falls, Cratr Fa, Dwy Sat, Ehidna Spik and Farful Drop. Th amp sits hav walking traks linking thm to various othr amp sits.

5 2. In a stat forst thr ar six amp sits: Angl Vally, Booming Falls, Cratr Fa, Dwy Sat, Ehidna Spik and Farful Drop. Th amp sits hav walking traks linking thm to various othr amp sits. Th matrix rprsnting th walks and amp sits is givn as shown. A B C D E F A1 é ù B C D E F ë û a In th matrix how many loops ar thr? In th matrix how many multipl dgs ar thr? Stat th dgr of vrtx A. d Stat th dgr of vrtx D. Rprsnt th matrix as a planar graph. f From th graph draw any tr. Qustions 3 to 6 ar asd upon information otaind from answrs to qustion 2, spifially th drawing of th graph in Two of th paths from A to D hav n losd du to wash- outs, and th loop out from A has n losd du to th nxt wk ing th rding sason for th rar alino ground- dwlling arthworm. Th rangrs hav providd th following (inomplt) matrix of distans (km) twn th amp sits. A B C D E F a d Aé ù B C D 0-10 E 0 7 F ë 0 û Why ar th zros running through th diagonal? What do th dashs ( ) indiat in trms of pathways twn amp sits? Fill in th lank spas in th matrix. Was it nssary to omplt th matrix? Why or why not? Rdraw th ntwork with th losd paths rmovd, ut inlud th distans givn in th matrix.

6 4. A group of mmrs from TLTHTB Fitnss Clu wish to omplt a run whih inluds ah of th paths ovr thr days. a Starting at A, plan a ours for th runnrs so that ah path is ovrd xatly on. What typ of path is this? Is it possil to mt th rquirmnts of th group and also hav th start and finish at A? Why or why not? d If your answr to part was no: i ould you suggst on path whih th rangrs must add to nal th iruit to ompltd? ii would this still nal th ntwork to planar? If th graph was no longr planar, what would this man aout th atual paths? 5. Anothr group, Th Walkrs Clu, is planning to amp at amp sit F. Thy wish to visit ah amp sit during thir 3- day stay and rturn to thir as amp ah night y us. Th total distan that thy plan to ovr ovr th thr days is 47 km. Assuming that th walks gt progrssivly longr and that no walk is ovr 20 km: a draw and larly lal ah of th daily walks draw th minimum spanning tr and alulat th total distan of this tr dtrmin what rquirmnt of th walkrs prvnts th minimum spanning tr ing utilisd. 6. Rylal matrials ar to olltd from housholds in a part of a partiular suur. Th ntwork rprsnts this ara, whr th numrd dots ar strt intrstions and th dgs ar th strts. Th numrs (in lu) indiat th lngths (in mtrs) of th strts twn intrstions. a d f g Dos this ntwork rprsnt a planar graph? Explain your answr. Vrify Eulr s formula for this ntwork. Rprsnt th ntwork as a matrix. Th olltors wish to travl along ah strt on only, in ordr to kp travl distan to a minimum. What sort of path is ndd? Dtrmin this path. As wll as th ritrion mntiond in part d aov, thy would also lik to start and finish at th sam intrstion. What sort of path is this? If this is possil, dtrmin this path. If it is not, what would nd to happn for it to possil? Th Transport Dpartmnt has plad traffi dnsity monitors at ah intrstion. Dtrmin a rout that a dpartmntal offir ould tak in ordr for hr to ollt all th monitors without rtraing a strt. What would this path alld? A taxi drivr wants to transport a ustomr from 1 to 10. Dtrmin th shortst distan for him to travl twn intrstions 1 and 10 and indiat this path.

7 7. An airfright ompany oprats out of Mlourn (M) to Glong (G), Ballarat (B), Castlmain (C), Symour (S), Mansfild (F) and Warragul (W). A planar graph of th ompany s flight ntwork is outlind, with distans shown in kilomtrs. Not that som flight paths, suh as Mlourn to Warragul, ar prohiitd so thy do not appar on th ntwork. a d f Vrify Eulr s formula for this ntwork. Rprsnt th ntwork as a matrix. Th ompany wishs to fly to all six of ths satllit towns and rturn to Mlourn. Mathmatially, what is this typ of path alld? Dtrmin th two possil flight paths for this. Draw a spanning tr for this ntwork. Now dtrmin th minimum spanning tr and th total distan rprsntd. Commnt on th usfulnss of this for th ompany. Us Dijkstra s algorithm to dtrmin th shortst distan twn Glong and Mansfild

Topi rviw answrs Multipl hoi 1 E 2 C 3 D 4 D 5 C 6 C 7 C 8 B 9 C 10 A 11 C Short answr 1 a V = {A, B,

G), (G, H)} A B C D E F G H A é 0 1 1 0 0 0 1 1ù B 1 0 1 1 1 0 0 0 C 1 1 0 0 0 0 0 0 D 0 1 0 0 1 0 0

8 Topi rviw answrs Multipl hoi 1 E 2 C 3 D 4 D 5 C 6 C 7 C 8 B 9 C 10 A 11 C Short answr 1 a V = {A, B, C, D, E, F, G, H} E = {(A, B), (A, C), (A, G), (A, H), (B, C), (B, D), (B, E), (D, E), (E, F), (F, G), (G, H)} A B C D E F G H A é ù B C D E F G H ë û 2 Diagram may somwhat diffrnt from figur. Sum of dgrs = 16 3 a V = 5, E = 6, F = 3, thus 5 = A B C D B F D E F A 5 A B C D E F A or A F E D C B A 6 a 7 a So thy ould start and finish at th sam hous. M D P B M K P B K D M

8 1 2 3 4 5 6 7 8 9 9 Distan = 199 km Extndd rspons 1 a

E3 E4 G E3 PT F E2 Othrs possil d Minimum spanning tr;

9 Distan = 199 km Extndd rspons 1 a Connt Flowr d to Lak An Eulr iruit: E2 K L F K E1 OT E4 L E3 E4 G E3 PT F E2 Othrs possil d Minimum spanning tr; distan = 400 m E2 K L E4, distan = 185 m 2 a d 6 f Answrs will vary. 3 a With th loops rmovd thr ar no paths laving any amp sit and oming dirtly ak to th amp sit. Thr ar no walks twn ths amp sits. A B C D E F Aé ù B C D E Fë û

10 d No, th matrix is symmtrial aout th zros. 4 a A E F D A C D B C F (Othrs ar possil ut thy must start at ithr A or F.) Eulr path No, thr ar two vrtis of odd dgr. d i A path from A to F ii Ys Th walks would most likly hav intrstions. 5 a Th rquirmnt that thy must rturn to amp and ovr no mor than 20 km pr day. 6 a Ys. No dgs ross. V = 10, E = 15, F = 7, so 10 = é ù ë û d Eulr path. Svral options ar possil, ut th path must start at vrtx 2 or 5 and finish at th othr (as thr ar two vrtis of odd dgr) Eulr iruit. Not possil as all vrtis must of an vn dgr. Anothr strt nds to addd twn intrstions 2 and 5. f On possil path is A Hamiltonian path g Shortst path is and is 450 m.

11 7 a V = 7, E = 11, F = 6, so 7 = é ù ë û A Hamiltonian yl M G W F S C B M and M B C S F W G M d Svral possiilitis Total distan = 510 km Of littl us aus thy nd to start and finish in Mlourn and flight paths would aktrak. f 240 km

Final Exam Solutions

CS 2 Advancd Data Structurs and Algorithms Final Exam Solutions Jonathan Turnr /8/20. (0 points) Suppos that r is a root of som tr in a Fionacci hap. Assum that just for a dltmin opration, r has no childrn