To determine which of the statements is false, we will analyze each statement one by one.

Statement 1: Every minimum spanning tree of G must contain emin.
To check the validity of this statement, we need to understand the concept of a minimum spanning tree (MST). An MST is a tree that spans all the vertices of a graph with the minimum possible total edge weight. Since emin is the edge with the minimum weight, it is reasonable to assume that it will be included in every MST. Therefore, statement 1 is true.

Statement 2: G has a unique minimum-spanning tree.
To verify the validity of this statement, we need to consider the conditions for a graph to have a unique MST. In an undirected connected graph with distinct edge weights, the MST is unique if and only if there are no two edges with the same weight. Since the given graph has distinct edge weights, it is possible for G to have multiple MSTs. Therefore, statement 2 is false.

Statement 3: No minimum spanning tree contains emax.
To evaluate the truthfulness of this statement, we need to consider the definition of emax, which is the edge with the maximum weight. In an MST, the goal is to minimize the total weight, so it is unlikely for emax to be included in the MST. Therefore, statement 3 is true.

Statement 4: If emax is in a minimum spanning tree, then its removal must disconnect G.
To determine the accuracy of this statement, we need to understand the impact of removing emax from an MST. If emax is included in the MST, its removal will not necessarily disconnect G. The MST is a tree that spans all the vertices, so removing one edge will not disconnect the graph unless it is a crucial bridge between disconnected components. Therefore, statement 4 is false.

In conclusion, the false statement is statement 2: G has a unique minimum-spanning tree.

Question

To determine which of the statements is false, we will analyze each statement one by one.

Statement 1: Every minimum spanning tree of G must contain emin.
To check the validity of this statement, we need to understand the concept of a minimum spanning tree (MST). An MST is a tree that spans all the vertices of a graph with the minimum possible total edge weight. Since emin is the edge with the minimum weight, it is reasonable to assume that it will be included in every MST. Therefore, statement 1 is true.

Statement 2: G has a unique minimum-spanning tree.
To verify the validity of this statement, we need to consider the conditions for a graph to have a unique MST. In an undirected connected graph with distinct edge weights, the MST is unique if and only if there are no two edges with the same weight. Since the given graph has distinct edge weights, it is possible for G to have multiple MSTs. Therefore, statement 2 is false.

Statement 3: No minimum spanning tree contains emax.
To evaluate the truthfulness of this statement, we need to consider the definition of emax, which is the edge with the maximum weight. In an MST, the goal is to minimize the total weight, so it is unlikely for emax to be included in the MST. Therefore, statement 3 is true.

Statement 4: If emax is in a minimum spanning tree, then its removal must disconnect G.
To determine the accuracy of this statement, we need to understand the impact of removing emax from an MST. If emax is included in the MST, its removal will not necessarily disconnect G. The MST is a tree that spans all the vertices, so removing one edge will not disconnect the graph unless it is a crucial bridge between disconnected components. Therefore, statement 4 is false.

In conclusion, the false statement is statement 2: G has a unique minimum-spanning tree.

Knowee AI · Accepted Answer

To determine which of the statements is false, we will analyze each statement one by one.

Statement 1: Every minimum spanning tree of G must contain emin.
To check the validity of this statement, we need to understand the concept of a minimum spanning tree (MST). An MST is a tree that spans all the vertices of a graph with the minimum possible total edge weight. Since emin is the edge with the minimum weight, it is reasonable to assume that it will be included in every MST. Therefore, statement 1 is true.

Statement 2: G has a unique minimum-spanning tree.
To verify the validity of this statement, we need to consider the conditions for a graph to have a unique MST. In an undirected connected graph with distinct edge weights, the MST is unique if and only if there are no two edges with the same weight. Since the given graph has distinct edge weights, it is possible for G to have multiple MSTs. Therefore, statement 2 is false.

Statement 3: No minimum spanning tree contains emax.
To evaluate the truthfulness of this statement, we need to consider the definition of emax, which is the edge with the maximum weight. In an MST, the goal is to minimize the total weight, so it is unlikely for emax to be included in the MST. Therefore, statement 3 is true.

Statement 4: If emax is in a minimum spanning tree, then its removal must disconnect G.
To determine the accuracy of this statement, we need to understand the impact of removing emax from an MST. If emax is included in the MST, its removal will not necessarily disconnect G. The MST is a tree that spans all the vertices, so removing one edge will not disconnect the graph unless it is a crucial bridge between disconnected components. Therefore, statement 4 is false.

In conclusion, the false statement is statement 2: G has a unique minimum-spanning tree.

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