Question & Answers >Assignment CLO 3: Graphs and Trees Discrete Mathematics (CII1G3) Second Term 202

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NIM: 1301204188

Name: Akmal Muhamad Firdaus

Class: IF4401

Assignment CLO 3: Graphs and Trees Discrete Mathematics (CII1G3) Second Term 2020-2021 Instructions: 1. This assignment is due Friday June 4, 2021 at 5:00 p.m.. Please submit your work to the correspond-

ing submission slot in LMS CeLOE. You need to submit a readable .pdf file of this assignment to the provided submission slot in CeLOE. You can contact your class instructor for more detailed information.

Please make sure that your file size do not exceed the maximum file size allowed.

2. Please upload your assignment to the LMS CeLOE under the file name: A3-cstudent ID>.pdf,

for example: A3-1301208888.pdf.

3. You may submit this assignment in one the following form:

a.You print this assignment and write your answer using HB/2B pencil or pen with blue/black ink (handwritten answer). You may add additional A4-sized papers. Afterwards you submit the scan/photograph of this assignment. b.You use a .pdf editing tools and write your answer directly using blue/black colored writing. c.You copy the problem from this assignment to a text/word processing program and type your answer neatly. d.You rewrite the problem from this assignment in an A4-sized paper and submit the scan/photograph of your work.

4. All problems in this assignment are adapted from the textbooks. The problems are written in English.

If you are a student in a regular class, you may answer the problems in Bahasa Indonesia. However, if you are a student in international class, your answers must be written in English—otherwise your assignment will not be graded. You may ask your class instructor or teaching assistant for helping you understanding the problem, but you should not ask them to give the solution of any problem.

5. Be neat and write legibly. You will be graded not only on the correctness of your answers, but also on

the clarity with which you express them.

6. This assignment consists of 10 problems and each problem is worth 10 points.

7. Please retain yourself from copying answers from elsewhere without understanding the steps. Such an

attitude will not enhance your knowledge. This assignment is an individual evaluation.

8. Important: late submission without reasonable explanation will not be graded.

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Problem 1 A simple undirected graph is an undirected graph with no parallel edges or loops. A h-regular graph is a simple connected graph whose every vertex has degree h. (a). [5 points] Is there any †-regular graph with seven vertices? If so, draw such a graph; otherwise, explain your answer. (b). [5 points] Is there a h-regular graph with h ‡ 2 vertices and fi2 edges? If so, draw such a graph; otherwise, explain your answer.

ANSWER:

A. Tidak ada, dengan teorema jabat tangan maka ∑vcV deg(v) = ∑vcV 5 = 5(7) = 35 = 2 |E|. maka |E| = 17,5 Z. B. Ada, misalkan G = (V,E) adalah graf K-regular dengan k+2 simpul dan 12 sisi, dengan teorema Jabat tangan. ∑vcV deg(v) = 2|E| ∑vcV k = 2.12 (k+2)k=24 K2+2k-24 = 0 (k+6)(k-4) = 0 Maka, kita memiliki k = -6/k = 4. Karena K harus bil. Positif, maka k = 4. Maka sebuah graf 4-regular dengan 6 sudut memiliki 1/2 ∑vcV 4 = ½(4.6) = 12 sisi.

page 2 of 14

Name: Akmal Muhamad Firdaus

NIM: 1301204188

Class: IF4401

Problem 2 Suppose G is an undirected graph depicted in Figure 1.

Figure 1: An undirected multigraph G.

(a). [5 points] Construct the adjacency matrix of G (that is, the matrix AC). (b). [5 points] Determine the number of different paths of length three from (i) a to b. (ii) b to c.

(Hint: calculate A3C.) ANSWER:

0 2 A. AG = [ 1 0

2 0 1 1

1 1 0 0

0 1 ] 0 0

0 2 B. A2G = [ 1 0

2 0 1 1

1 1 0 0

0 0 1 2 ][ 0 1 0 0

5 1 A3G = [ 2 2

1 6 2 0

2 2 2 1

2 0 0 2 ][ 1 1 1 0

2 0 1 1

1 1 0 0 2 0 1 1

0 5 1 1 ]=[ 0 2 0 2 1 1 0 0

1 6 2 0

2 2 2 1

2 0 ] 1 1

0 4 1 14 ]=[ 0 6 0 1

14 4 7 6

6 7 4 2

1 6 ] 2 0

Maka : (i) Ada 14 jalan berbeda dari a ke b. (ii) Ada 7 jalan berbeda dari b ke c

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Problem 3 In the movie “Good Will Hunting” (see http://www.imdb.com/title/tt0119217/), the main character Will Hunting secretly solves a blackboard problem which had been assigned as a challenge to a class1. Two of these problems are now become the following problems of this assignment. The problems involve the graph G in Figure 2.

Figure 2: Graph G in “Good Will Hunting” movie.

(a). [4 points] Find AC, i.e., the adjacency matrix of G. (b). [4 points] Find the matrix giving the number of paths of length 3 in G. (Several examples of paths of length 3 are: (efi, e2, e3), (efi, e3, e2), and (e4, e5, e3).) (c). [2 points] Write the name of the actor who played Will Hunting. (Note: you does not have to watch the entire movie.)

ANSWER:

0 1 0 1 1 0 2 1 A. AG = [ ] 0 2 0 0 1 1 0 0 B. Matrix dengan panjang jalan 3 dalam G setara dengan A3G 0 1 0 1 0 1 0 1 2 1 2 1 1 0 2 1 1 0 2 1 1 6 0 1 A2G = [ ][ ]=[ ] 0 2 0 0 0 2 0 0 2 0 4 2 1 1 0 0 1 1 0 0 1 1 2 2 2 1 2 1 0 1 0 1 2 7 2 3 7 2 12 7 1 6 0 1 1 0 2 1 A3G = [ ][ ]=[ ] 0 2 0 0 2 12 0 2 2 0 4 2 1 1 2 2 1 1 0 0 3 7 2 2 C. Will Hunting: Matthew Paige Damon

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Name: Akmal Muhamad Firdaus

NIM: 1301204188

Class: IF4401

Problem 4 A region consists of nine villages labelled as a, b, c, . . . , s. The road network connecting them can be represented using graph G in Figure 3. The vertices represent the villages while the edges denote the road between two adjacent villages. Until recently, all roads are not yet paved.

Figure 3: Graph G representing a road network among nine villages in a region.

One day, the government plans to pave all roads in Figure 3. Nevertheless, the government can only employ one construction company. To attain efficiency, the company aim to pave all roads continuously without stopping (unless all roads have been paved). In addition, a road connecting two cities must be paved once and the company do not want to traverse the road without paving it. (a).Explain the solutions of the following problems clearly. (i) [3 points] Is it possible to pave the roads in Figure 3 continuously without traversing the same road twice if the construction must start and end at the same village? If so, provide the path; if not, explain your reasoning. (ii) [4 points] Is it possible to pave the roads in Figure 3 continuously without traversing the same road twice if the construction can start and end at different villages? If so, provide the path; if not, explain your reasoning. (b). [3 points] A government representative plans to visit all villages in the region. He plans to begin his journey from village a. To save time he plans to visit all villages exactly once and return to the initial village (village a). Is this plan possible? If so, provide the path of his journey; otherwise explain your reasoning.

ANSWER:

A. (i) Tidak, karena graf tersebut tidak memiliki sirkuit Euler. Karena memiliki 2 simpul berderajat ganjil (ii) Iya, karena graf tersebut memiliki jalur Euler, dan memiliki 2 simpul berderajat ganjil Salah satu jalurnya (a, c, f, g, h, i, e, h ,d ,g , c, d, e, b, d, a, b) B. Iya, karena grafnya memiliki sirkuit Hamilton, contoh salah satu jalurnya (a, b, e, i, h, g , f, c, d, a)

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Problem 5 The graph K5 is a simple undirected graph with five vertices in which every pair of different vertices is connected by an edge. This graph is illustrated in Figure 4.

Figure 4: Graph K5.

(a). [5 points] Determine the number of different spanning subgraphs of K5 that has four edges (i.e., the subgraph of K5 with four edges which contains all the five vertices). An example of this subgraph is G = (V, E) where V = (a, b, c, d, e} and E = ((a, b} , (b, c} , (c, d} , (d, e}}. Express your result as an integer! (b). [5 points] Draw three non-isomorphic spanning trees of K5. Explain why these trees are not isomorphic.

ANSWER:

A. Misalkan K5 = (VK5, EK5) dan G = (V, E) adalah subgraph K5 yang memiliki 4 sudut. Maka V = Vk5 = {a, b, c, d, e} dan E EK5 dengan |E| = 4. Jadi angka perbedaan subgraph K5 setara dengan perbedaan 10 subset EK5 yang memiliki 4 elemen. Karena Ek5 = 10, maka perbedaan subset EK5 setara dengan ( ) = 4 210.

B.

Ketiganya tidak isomorfik karena T1 ≠ T2. Karena T1 tidak memiliki vertex berderajat 4 T1 ≠ T3. Kareen T1 tidak memiliki vertex berderajat 3 T2 ≠ T3. Karena T3 tidak memiliki vertex berderajat 4

page 6 of 14

Name: Akmal Muhamad Firdaus

NIM: 1301204188

Class: IF4401

Problem 6 During the semester break, Ahsan works as a tutor for competitive programming boot camp. The boot camp is conducted in three weeks during the holiday. The participants of the boot camp are divided into six classes, namely class A, B, C, and so on until class F. Ahsan is in charge to determine the schedule for the lab works of each class. Suppose the training camp is conducted from Monday to Friday and the schedule for theory session for each class is illustrated in Table 1.

Time slot Class A OF : OO — O9 : OO 5 O9 : OO — fifi : OO 5 fifi : OO — fi3 : OO — fi3 : OO — fi† : OO — fi† : OO — fiF : OO —

Class B — — — 5 5

Class C — 5 5 — —

Class D — — 5 5 —

Class E 5 — 5 — —

Class F — — 5 — 5

Tabel 1: Class schedule for theory session.

In this problem, your task is to help Ahsan in determining the minimum number of slots required for the lab works session of each class. The lab works session can be combined if it involves less than three classes. We cannot combine the lab works session for three classes or more. The problem can be solved using graph colorization approach. We use the graph G = (V, E) where V = (A, B, C, D, E, F} denotes the classes and two vertices are adjacent if and only if two classes share the same schedule for the theory session. (a). [5 points] Complete the following simple graph G = (V, E) in Figure 5 where V = (A, B, C, D, E, F} represents the classes and two vertices are adjacent if and only if two classes share the same schedule for the theory session. For example, since class A and class E both share the theory session at OF : OO — O9 : OO, there is an edge connecting the vertices A and E. In addition, since class A and class C both share the theory session at O9 : OO — fifi : OO, there is an edge connecting the vertices A and C.

Figure 5: Graph G for scheduling problem.

page 7 of 14

(b). [5 points] Use Welsh-Powell algorithm to perform vertex coloring on the graph obtained in part (a). Describe the process in tabular format. What is the chromatic number of the graph G? (The chromatic number of G is the minimum number of colors required to color all vertices of G so that no two adjacent vertices use the same color.) Draw the resulting colored graph in Figure 6. How many different lab works slots required in the boot camp? Provide one possible scheduling of the lab works!

Figure 6: The vertex coloring result of graph G. ANSWER:

Vertex Derajat Warna X(G) = 4

page 8 of 14

A 2 Kuning

B 2 Hijau

C 4 Merah

D 4 Kuning

E 4 Hijau

F 4 Biru

Name: Akmal Muhamad Firdaus

NIM: 1301204188

Class: IF4401

Problem 7 Salvador works as a project manager. He leads a team of programmers that consists of six people: Alice, Bob, Carlos, David, Emma, and Fiona. Salvador wants them to work together, but unfortunately some of his team members hate each other. Salvador doesn’t want his team members work individually because it can slow down the projects. He plans to divide his team into smaller groups. From the most recent information, Salvador knows that: 1. Alice hates everyone except David, 2. Bob hates everyone except Carlos and Emma, 3. Carlos hates Alice and Emma, 4.

5.

David hates Bob and Emma, Emma hates everyone except Bob,

6. Fiona hates everyone except Carlos and David, Two people who hate each other cannot be grouped together because it can hamper the project. (a). [5 points] Complete the simple graph G = (V, E) in Figure 7 with V = (a, b, . . . , f} that represents the hatred among the programmers, where a represents Alice, b represents Bob, and so on. Two vertices are adjacent if and only if two people hate each other. We assume that hatred is symmetric, i.e., if person x hates person y, then person y also hates person x. For example, since David hates Bob and Emma, we have edges (b, d} and (d, e}.

Figure 7: Graph representing hatred among six programmers.

page 9 of 14

(b). [5 points] Use Welsh-Powell algorithm to perform vertex coloring on the graph obtained in part (a).

Is it possible to use each color at least twice? (That is, for each color, there are at least two vertices of the same color.) Describe the process in tabular format. Draw the resulting colored graph in Figure 8. What is the chromatic number of the graph G? (The chromatic number of G is the minimum number of colors required to color all vertices of G so that no two adjacent vertices use the same color.) Is it possible to group the programmers according to the above-mentioned conditions and no programmer works individually? If so, provide an example of the grouping (determine the member of each group); otherwise, explain why the grouping is impossible.

Figure 8: The vertex coloring result of graph G in Figure 7.

ANWSER:

X(G) = 3, dan Setiap warna dapat menggunakan warna dua kali Vertex Derajat Warna

page 10 of 14

A 4 Merah

B 3 Hijau

C 2 Biru

D 2 Merah

E 4 Hijau

F 3 Biru

Name: Akmal Muhamad Firdaus:

NIM: 1301204188

Class: IF4401

Problem 8 Suppose G and H are two graphs in Figure 9.

Figure 9: Graph G (left) and graph H (right).

(a). [2 points] Construct AC (the adjacency matrix of G) and AH (the adjacency matrix of H), each with row and column labels a, b, . . . , ƒ (for G) and p, q, . . . , u (for H), respectively. Write the matrices carefully with correct notation. No partial credits for wrong answer. (b). [2 points] Construct the adjacency lists of G and H. Write the lists carefully. No partial credits for wrong answer. (c). [3 points] Determine whether graph H in Figure 9 is planar or not (i.e., whether H can be drawn without edges crossing or not). If so, redraw H so that no edges cross; otherwise, explain your answer mathematically. (d). [3 points] Determine whether G and H are isomorphic. Exhibit an isomorphism or provide a rigorous argument that none exist. If the graphs are isomorphic, show that we can rearrange the vertices of H so that AC = AtH , where AtH is the adjacency matrix of H with a particular rows and columns relabeling.

ANWSER:

0 0 0 0 0 0 0 0 0 A. AG = 1 1 1 1 1 1 [1 1 1 B. G = vertex

1 1 1 0 0 0

1 1 0 1 1 1 1 1 0 AH = 0 0 1 0 0 0 [1 0 0] tetangga

1 0 1 1 0 0

0 1 0 0 1 1

1 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0] H = Vertex

tetangga

page 11 of 14

ANWSER:

C. H adalah planar, contohnya

D. Grafik G dan H bukan isomorfik

page 12 of 14

Name: Akmal Muhamad Firdaus

NIM: 1301204188

Class: IF4401

Problem 9 A rooted tree is called a full m-ary tree if its root and internal vertices have exactly m children (each of them has precisely m children). If m = 3, then the tree is called as a full ternary tree. The level of a vertex in a full m-ary tree is the number of edges on a unique path between such vertex and a root. The height of a rooted tree is the largest level that may exist in such tree. An m-ary rooted tree with h levels is called a balanced m-ary tree if all of its leaves are at level h or h — fi. (a). [5 points] A leaf in a rooted tree is a node that has no child. Draw (if any) an example of a full and balanced ternary tree of height 2 with five leaves. (b). [5 points] Determine the height of a full and balanced ternary tree that has thirteen leaves.

A NWSER :

A.

B. H = 3

page 13 of 14

Problem 10 A computer network in a room consists of seven computers labelled as a, b, . . . , and g. The network is illustrated in Figure 10. The vertices represent the computers and the edges are the optical cables that connect them. The length (in feet) of each cable that connects two computers is depicted as a number assigned to it. For instance, the length of the cable that connects computer a and b is 2 feet.

Figure 10: Graph for computer network.

(a). [5 points] Use Prim’s algorithm to find the configuration. Describe the iterations in tabular format, calculate the total cables’ length, and draw the resulting network. (b). [5 points] Use Kruskal’s algorithm to find the configuration. Describe the iterations in tabular format, calculate the total cables’ length, and draw the resulting network.

ANWSER:

A. No Ujung Panjang

1 {a, d} 1

2 {a, b} 2

3 {c, d} 2

4 {d, g} 4

5 {f, g} 1

6 {e, g} 6

No 1 2 3 4 Ujung {a, d} {f, g} {c, d} {a, b} Panajng 1 1 2 2 Total panjang kabel 1+1+2+2+4+6 = 16. Sama dengan gambar A

5 {d, g} 4

6 {e, g} 6

Total panjang kabel 1+2+2+4+1+6 =16.

B.

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