# Graph Theory

 Question 1
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 ______
 A 929 B 254 C 639 D 879
GATE CSE 2021 SET-2   Discrete Mathematics
Question 1 Explanation:
 Question 2
Consider the following directed graph:

Which of the following is/are correct about the graph?
[MSQ]
 A The graph does not have a topological order B A depth-first traversal starting at vertex S classifies three directed edges as back edges C The graph does not have a strongly connected component D For each pair of vertices u and v, there is a directed path from u to v
GATE CSE 2021 SET-2   Discrete Mathematics
Question 2 Explanation:
 Question 3
An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components.
Let T be a DFS tree obtained by doing DFS in a connected undirected graph G.
Which of the following options is/are correct?
 A Root of T can never be an articulation point in G. B Root of T is an articulation point in G if and only if it has 2 or more children. C A leaf of T can be an articulation point in G. D If u is an articulation point in G such that x is an ancestor of u in T and y is a descendent of u in T, then all paths from x to y in G must pass through u.
GATE CSE 2021 SET-1   Discrete Mathematics
Question 3 Explanation:
 Question 4
G is an undirected graph with vertex set {v1, v2, v3, v4, v5, v6, v7} and edge set {v1v2, v1v3, v1v4 ,v2v4, v2v5, v3v4, v4v5, v4v6, v5v6, v6v7 }. A breadth first search of the graph is performed with v1 as the root node. Which of the following is a tree edge?
 A v2v4 B v1v4 C v4v5 D v3v4
ISRO CSE 2020   Discrete Mathematics
Question 4 Explanation:
 Question 5
Graph G is obtained by adding vertex s to $K_{3,4}$ and making s adjacent to every vertex of $K_{3,4}$. The minimum number of colours required to edge-colour G is _______
 A 4 B 5 C 6 D 7
GATE CSE 2020   Discrete Mathematics
Question 5 Explanation:
 Question 6
Let G be an undirected complete graph on n vertices, where n$\gt$2. Then, the number of different Hamiltonian cycles in G is equal to
 A n! B (n-1)! C 1 D $\frac{(n-1)!}{2}$
GATE CSE 2019   Discrete Mathematics
Question 6 Explanation:
 Question 7
The number of edges in a regular graph of degree: d and n vertices is:
 A maximum of n and d B n+d C nd D nd/2
ISRO CSE 2018   Discrete Mathematics
Question 7 Explanation:
 Question 8
Which of the following is application of Breath First Search on the graph?
 A Finding diameter of the graph B Finding bipartite graph C Both (A) and (B) D None of the above
ISRO CSE 2018   Discrete Mathematics
Question 8 Explanation:
 Question 9
Let G be a graph with 100! vertices, with each vertex labelled by a distinct permutation of the numbers 1,2,...,100. There is an edge between vertices u and v if and only if the label of u can be obtained by swapping two adjacent numbers in the label of v. Let y denote the degree of a vertex in G, and z denote the number of connected components in G. Then, y+ 10z = _____.
 A 83 B 45 C 201 D 109
GATE CSE 2018   Discrete Mathematics
Question 9 Explanation:
 Question 10
Let G be a simple undirected graph. Let TD be a depth first search tree of G. Let TB be a breadth first search tree of G. Consider the following statements.

(I) No edge of G is a cross edge with respect to TD. (A cross edge in G is between two nodes neither of which is an ancestor of the other in TD.)
(II) For every edge (u,v) of G, if u is at depth i and v is at depth j in TB, then |i-j|=1.

Which of the statements above must necessarily be true?
 A I only B II only C Both I and II D Neither I nor II
GATE CSE 2018   Discrete Mathematics
Question 10 Explanation:

There are 10 questions to complete.

### 4 thoughts on “Graph Theory”

1. Question 6 is of group theory.