Minimum Spanning Tree

Question 1
Let G(V, E) be a directed graph, where V = \{1, 2, 3, 4, 5 \} is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A.
A[i][j]= \left \{ \begin{matrix} 1 &1 \leq j \leq i \leq 5 \\ 0& otherwise \end{matrix} \right.
A[i][j]=1 indicates a directed edge from node i to node j . A directed spanning tree of G , rooted at r \in V , is defined as a subgraph T of G such that the undirected version of T is a tree, and T contains a directed path from r to every other vertex in V . The number of such directed spanning trees rooted at vertex 5 is ____
A
24
B
36
C
12
D
6
GATE CSE 2022   Algorithm
Question 2
Consider a simple undirected weighted graph G, all of whose edge weights are distinct. Which of the following statements about the minimum spanning trees of G is/are TRUE?
MSQ
A
The edge with the second smallest weight is always part of any minimum spanning tree of G .
B
One or both of the edges with the third smallest and the fourth smallest weights are part of any minimum spanning tree of G .
C
Suppose S\subseteq V be such that S\neq \phi and S\neq V . Consider the edge with the minimum weight such that one of its vertices is in S and the other in V \ S . Such an edge will always be part of any minimum spanning tree of G .
D
G can have multiple minimum spanning trees.
GATE CSE 2022   Algorithm
Question 3
Let G be a connected undirected weighted graph. Consider the following two statements.

S1: There exists a minimum weight edge in G which is present in every minimum spanning tree of G.
S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.

Which one of the following options is correct?
A
Both S1 and S2 are true
B
S1 is true and S2 is false
C
S1 is false and S2 is true
D
Both S1 and S2 are false
GATE CSE 2021 SET-2   Algorithm
Question 4
Consider the following undirected graph with edge weights as shown:

The number of minimum-weight spanning trees of the graph is __________
A
4
B
6
C
5
D
3
GATE CSE 2021 SET-1   Algorithm
Question 5
Consider a graph G=(V,E), where V=\{v_1,v_2,...,v_{100}\}, E=\{(v_i,v_j)|1\leq i \lt j\leq 100\}, and weight of the edge (v_i,v_j)\; is \; |i-j|. The weight of minimum spanning tree of G is _________
A
100
B
99
C
199
D
90
GATE CSE 2020   Algorithm
Question 6
Let G=(V,E) be a weighted undirected graph and let T be a Minimum Spanning Tree (MST) of G maintained using adjacency lists. Suppose a new weighed edge (u,v)\in V \times V is added to G. The worst case time complexity of determining if T is still an MST of the resultant graph is
A
\Theta (|E|+|V|)
B
\Theta (|E||V|)
C
\Theta (|E|\log |V|)
D
\Theta (|V|)
GATE CSE 2020   Algorithm
Question 7
Let G be any connection, weighted, undirected graph:

I. G has a unique minimum spanning tree if no two edges of G have the same weight.
II. G has a unique minimum spanning tree if, for every cut of G, there is a unique minimum weight edge crossing the cut.

Which of the above two statements is/are TRUE?
A
I only
B
II only
C
Both I and II
D
Neither I nor II
GATE CSE 2019   Algorithm
Question 8
Consider the following undirected graph G:

Choose a value for x that will maximize the number of minimum weight spanning trees (MWSTs) of G. The number of MWSTs of G for this value of x is ______.
A
3
B
4
C
5
D
1
GATE CSE 2018   Algorithm
Question 9
Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive and distinct. Consider the following statements:

(I) Minimum spanning tree of G is always unique.
(II) Shortest path between any two vertices of G is always unique.

Which of the above statements is/are necessarily true?
A
(I) only
B
(II) only
C
Both (I) and (II)
D
Neither (I) nor (II)
GATE CSE 2017 SET-1   Algorithm
Question 10
G = (V,E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof G. Which of the following statements about the minimum spanning trees (MSTs) of G is/are TRUE?
I. If e is the lightest edge of some cycle in G, then every MST of G includes e
II. If e is the heaviest edge of some cycle in G, then every MST of G excludes e
A
I only
B
II only
C
both I and II
D
neither I nor II
GATE CSE 2016 SET-1   Algorithm
There are 10 questions to complete.

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