COMSATS University Islamabad
Department of Computer Science
Terminal Examination – Fall 2023
Class: (BS) CS
Course Code & Title: CSC307 Graph Theory
Time Allowed: 3 Hours
Instructors: Dr. Malik A. Kamran
Semester: 4th
Date: 01-01-2024
Max Marks: 50
Name:___________________________
Registration #_________________
Instructions
• Use black or blue ball-pen only. Markers/lead pencils are not allowed.
• All questions are self-explanatory and require no further explanations during exam time.
• Attempt all questions.
Question 1
CLO-1
Marks [3+2+1+2+1+3+2+2+2]
CLO1: Illustrate the basic terminology of graph theory including properties and special
cases for each type of graph, trees, and forest.
1. Write G = (V, E) definitions and draw each of the following:
a) Two trees T1 and T2 have 5 nodes each. Draw trees such that T1 has maximum number
of leaf and T2 has minimum number of leaf.
b) Afterwards, convert T1 to a wheel graph W and T2 to a cycle C (by adding edges).
c) Using W and C graphs created in (b), create a connected graph G by connecting center
of the W to any one of the nodes in C.
d) How many cut vertices are there in each of the five graphs created above (T1, T2, W, C,
G)?
e) Draw a spanning tree of the connected graph G (G was created in (c)).
f) Using eccentricity, find center of the graph G (created in (c)).
2. What are different sized walks in a P3 that can be reduced to a path of length zero.
3. How many vertices a graph contains if it has 12 edges, 3 vertices of degree 2 and other vertices
of degree 3.
4. Adjacency matrix provides paths of length one. If adjacency matrix is multiplied by itself, it
provides paths of length 2. Verify it for C4.
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Question 2
CLO-2
Marks [2+2+4]
CLO2: Interpret basic results related with Eulerian and Hamiltonian graphs.
1. Degree of the vertices can indicate existence of Eulerian walk and Eulerian circuit. Prove
with example graphs.
2. Find Hamiltonian and/or Eulerian path/circuit if exists in the following graph.
3. Are all 2-regular graphs bipartite? Can you convert C6 into a bipartite graph.
Question 3
CLO-3
Marks [4+4+3]
CLO3: Apply network flow algorithms to solve graph connectivity problems.
1. Show all possible changes in the number of components by removing a vertex or edge from
different types of graphs.
2. Capacity is shown on the following network. Find maximum flow in the following network
using Ford-Fulkerson algorithm.
3. Convert the following graph to a network. Show that flow is equivalent to matching in this
network (capacity of all links is 1)?
Page 2 of 3
Question 4
CLO-4
Marks [2+3+3+3+2]
CLO4: Solve a variety of real-world problems in computer science using appropriate
forms of graphs.
1. Several students will take summer classes. The students’ schedules are shown. Draw a graph
representing this situation, and find the smallest number of slots that they can be scheduled
in.
2. Show max matching and min vertex cover in the following bipartite graph B:
B=(G, X, Y), G=({1,2,3,4,5},14,15,25,34,35), X={1,2,3}, Y={4,5}
3. Show dominating set, edge cover, and edge coloring in the following graph:
4. Determine the clique number, independence number, and chromatic number of the
following graph:
5. Draw a Latin square of 3x3 using formula. Convert it to a doubly Stochastic matrix.
Page 3 of 3
COMSATS University Islamabad (CUI)
Department of Computer Science
Terminal Examination Fall - 2023
BSCS– IV (A&B) SEMESTER
Course: CSC307 – Graph Theory
Maximum Marks: 25
Dated: 7th Nov , 2023
Instructor: Dr. Malik A. Kamran
Time Allowed: 80 Minutes
1. [CLO-1] Illustrate the basic terminology of graph theory including properties and special cases for
each type of graph, trees, and forest.
1.1 Let G be a graph with degree sequence is 4, 3, 3, 2, 2?
a) Draw G and show incidence matrix of G.
b) Show all dominating sets and detached pairs in G.
1.2 Whether G1 and G2 are Isomorphic? (Prove using properties of two graphs to be
isomorphic).
G1
[1+1]
[2+2]
[3]
G2
1.3 Let G1 is an empty graph, G2 is having single node, and G3 has two components. Which of
them are tree or graph?
[3]
2. [CLO-2] Interpret basic results related with Eulerian and Hamiltonian graphs.
2.1 A graph G1 is defined as having single component that consist of a C6 and edges 27, 67.
1..1. Show Hamiltonian path and/or cycle if exists.
[2]
1..2. Show Eulerian path and/or cycle if exists.
[2]
2.2 Differentiate walk, path, cycle, trail, and circuit based on the following characteristics:
Repeating vertices, repeating edges, open, and closed.
[2]
2.3 For the following digraph:
a) Find Strongly connected and weakly connected components.
b) Draw Drev and Complement of D.
[2+2]
[1+2]
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