# Is the minimum spanning tree of a given weighted graph unique?

## Is the minimum spanning tree of a given weighted graph unique?

Note that we use minimum spanning tree as short for minimum weight spanning tree. Weight of MST is 4 + 8 + 7 + 9 + 2 + 4 + 1 + 2 = 37 • Note: MST is not unique: e.g. (b, c) can be exchanged with (a, h) 1 Page 2 The MST problem is considered one of the oldest, fundamental problems in graph algorithms.

### What makes a minimum spanning tree unique?

Uniqueness. If each edge has a distinct weight then there will be only one, unique minimum spanning tree. This is true in many realistic situations, such as the telecommunications company example above, where it’s unlikely any two paths have exactly the same cost. This generalizes to spanning forests as well.

What is the weight of minimum spanning tree?

A spanning tree with assigned weight less than or equal to the weight of every possible spanning tree of a weighted, connected and undirected graph G, it is called minimum spanning tree (MST). The weight of a spanning tree is the sum of all the weights assigned to each edge of the spanning tree.

How do you know if a graph has a unique MST?

You can prove whether it has a unique MST in O(E log(V)) . First find a minimum spanning tree with standard techniques. Go back to your original tree, and replace all weights with pairs of numbers, the original weight, and then 0 or 1 based on whether or not it is in the MST you found.

## Why the minimum spanning tree of a graph may not be unique?

The edge weights may be zero or negative. If the edge weights are all positive, it suffices to define the MST as the subgraph with minimal total weight that connects all the vertices. The edge weights are all different. If edges can have equal weights, the minimum spanning tree may not be unique.

### What is the weight of the minimum spanning tree of the given weighted graph?

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges(V – 1 ) of a connected, edge-weighted undirected graph G(V, E) that connects all the vertices together, without any cycles and with the minimum possible total edge weight.

How can I prove that any weighted connected graph with distinct weights has exactly one minimum spanning tree?

To make a MINIMUM spanning tree out of it, you have to remove the most expensive edge in the cycle. Because all the edges have different weights, the most expensive edge will be only one of its kind. If ‘e’ is the most expensive edge, then, you don’t get multiple MSTs.

How many distinct minimum spanning trees exist for the weighted graph below?

Minimum spanning tree will have weight 11, hence 6 is the correct answer.

## Does a graph with distinct edge weights have a unique minimum spanning tree?

If all edge weights in a connected graph G are distinct, then G has a unique minimum spanning tree.

### Is MST NP complete?

The fact that the k-MST problem is NP-complete for distance matrices in [RT], but polynomially solvable, when the distance matrix is in [RI], points out an interesting difference between these two at first sight similar problems.

Can a graph have more than one MST?

A graph can have more than one MST in the case where both trees have the same overall weight but different paths to complete the tree.

How many MST Can a graph have?

There is only one minimum spanning tree in the graph where the weights of vertices are different.