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


There are 5 questions to complete.

8 thoughts on “Minimum Spanning Tree”

  4. Hello Sir/Ma’am

    In Q.26, C option has repeated edges it should be like this….
    (C) (d—f),(a—b),(d—c),(b—f),(d—e)

    Reply

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