Minimum Spanning Tree

Question 1
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 2
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 3
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 4
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 5
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 6
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 7
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 8
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
Question 9
Let G be a complete undirected graph on 4 vertices, having 6 edges with weights being 1,2,3,4,5, and 6. The maximum possible weight that a minimum weight spanning tree of G can haveis .
A
6
B
7
C
8
D
5
GATE CSE 2016 SET-1   Algorithm
Question 10
Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increasedby the same value ,then which of the following statements is/are TRUE?
P: Minimum spanning tree of G does not change
Q: Shortest path between any pair of vertices does not change
A
P only
B
Q only
C
Neither P nor Q
D
Both P and Q
GATE CSE 2016 SET-1   Algorithm
There are 10 questions to complete.

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