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 ______

The sum of the quality-scores of all vertices on the graph shown above is ______

929 | |

254 | |

639 | |

879 |

Question 1 Explanation:

Question 2 |

Consider the following directed graph:

Which of the following is/are correct about the graph?

Which of the following is/are correct about the graph?

**[MSQ]**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 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?

Let T be a DFS tree obtained by doing DFS in a connected undirected graph G.

Which of the following options is/are correct?

Root of T can never be an articulation point in G. | |

Root of T is an articulation point in G if and only if it has 2 or more children. | |

A leaf of T can be an articulation point in G. | |

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. |

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?

v2v4 | |

v1v4 | |

v4v5 | |

v3v4 |

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 _______

4 | |

5 | |

6 | |

7 |

Question 5 Explanation:

Question 6 |

Let G be an undirected complete graph on n vertices, where n\gt2. Then, the number of different Hamiltonian cycles in G is equal to

n! | |

(n-1)! | |

1 | |

\frac{(n-1)!}{2} |

Question 6 Explanation:

Question 7 |

The number of edges in a regular graph of degree: d and n vertices is:

maximum of n and d | |

n+d | |

nd | |

nd/2 |

Question 7 Explanation:

Question 8 |

Which of the following is application of Breath First Search on the graph?

Finding diameter of the graph | |

Finding bipartite graph | |

Both (A) and (B) | |

None of the above |

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 = _____.

83 | |

45 | |

201 | |

109 |

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?

(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?

I only | |

II only | |

Both I and II | |

Neither I nor II |

Question 10 Explanation:

There are 10 questions to complete.

Question 6 is of group theory.

Please remove it from here.

Thank You Abhishek Chavle,

We have updated it.

Q 11 and 29 too

Thank You Abhishek Chavle,

We have updated it.