- Courses
- Tutorials
- Interview Prep
Topic Wise Quiz on Graph Theory
Question 1
Which of the following graphs is isomorphic to
A
B
C
D
Question 2
Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to
360
45
30
15
Question 3
In a directed acyclic graph with a source vertex s, the quality-score of a directed path is defined to be the product of the weights of the edges on the path. Further, for a vertex v other than s, the quality-score of v is defined to be the maximum among the quality-scores of all the paths from s to v. The quality-score of s is assumed to be 1.
The sum of the quality-scores of all vertices on the graph shown above is _______ .
929
81
729
1023
Question 4
Consider the following directed graph:
Which of the following is/are correct about the graph?
The graph does not have a topological order
A depth-first traversal starting at vertex S classifies three directed edges as back edges
The graph does not have a strongly connected component
For each pair of vertices u and v, there is a directed path from u to v
Question 5
Let T be a depth first search tree in an undirected graph G. Vertices u and n are leaves of this tree T. The degrees of both u and n in G are at least 2. which one of the following statements is true?
There must exist a vertex w adjacent to both u and n in G
There must exist a vertex w whose removal disconnects u and n in G
There must exist a cycle in G containing u and n
There must exist a cycle in G containing u and all its neighbours in G.
Question 6
Consider the DAG with Consider V = {1, 2, 3, 4, 5, 6}, shown below. Which of the following is NOT a topological ordering?
1 2 3 4 5 6
1 3 2 4 5 6
1 3 2 4 6 5
3 2 4 1 6 5
Question 7
Let G (V, E) be a directed graph with n vertices. A path from v
i
to v
j
in G is sequence of vertices (v
i
, v
i+1
, ......., v
j
) such that (vk, v
k+1
) ∈ E for all k in i through j - 1. A simple path is a path in which no vertex appears more than once. Let A be an n x n array initialized as follow
Consider the following algorithm.
for i = 1 to n
for j = 1 to n
for k = 1 to n
A [j , k] = max (A[j, k] (A[j, i] + A [i, k]);
Which of the following statements is necessarily true for all j and k after terminal of the above algorithm ?
A[j, k] ≤ n
If A[j, k] ≥ n - 1, then G has a Hamiltonian cycle
If there exists a path from j to k, A[j, k] contains the longest path lens from j to k
If there exists a path from j to k, every simple path from j to k contain most A[j, k] edges
Question 8
Which of the following statements is/are TRUE for undirected graphs?
P: Number of odd degree vertices is even.
Q: Sum of degrees of all vertices is even.
Neither P nor Q
Both P and Q
Q only
P only
Question 9
The line graph L(G) of a simple graph G is defined as follows: · There is exactly one vertex v(e) in L(G) for each edge e in G. · For any two edges e and e' in G, L(G) has an edge between v(e) and v(e'), if and only if e and e'are incident with the same vertex in G. Which of the following statements is/are TRUE?
(P) The line graph of a cycle is a cycle.
(Q) The line graph of a clique is a clique.
(R) The line graph of a planar graph is planar.
(S) The line graph of a tree is a tree.
P only
P and R only
R only
P, Q and S only
Question 10
The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?
(I) 7, 6, 5, 4, 4, 3, 2, 1
(II) 6, 6, 6, 6, 3, 3, 2, 2
(III) 7, 6, 6, 4, 4, 3, 2, 2
(IV) 8, 7, 7, 6, 4, 2, 1, 1
I and II
III and IV
IV only
II and IV
There are 36 questions to complete.