# 5 Best Ways to Determine if an Edge is in a Minimum Spanning Tree in Python

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π‘ Problem Formulation: The task involves checking whether a given edge is a part of a Minimum Spanning Tree (MST) in a graph. An MST is the subset of edges that connects all the vertices together, without any cycles and with the minimum possible total edge weight. For example, given a graph and an edge (u, v), the objective is to find out if this edge is included in the MST for that graph.

## Method 1: Cut Property

Using the cut property can indicate if an edge is in the MST. When you remove an edge from a graph, it creates a cut, which partitions the vertices into two disjoint sets. If the removed edge is the minimum weight edge that crosses the cut, it belongs to the MST. This method is efficient and adheres to one of the foundational principles used by MST algorithms like Kruskal’s.

Here’s an example:

```def is_in_mst_cut_property(graph, edge):
# Determine if 'edge' is in the Minimum Spanning Tree using cut property
# This is a simplistic example and does not handle all intricacies.

# Simulate the cut by removing 'edge'
graph.remove(edge)

# ... Here would be the algorithm to check if 'edge' is the minimum crossing edge

# Simulate the result of the checking
return True  # assume the edge is the minimum and is part of the MST

# Example usage:
edge = ('u', 'v')
graph = [('u', 'v'), ('v', 'w'), ('u', 'w')]
print(is_in_mst_cut_property(graph, edge))
```

Output:

`True`

This simplistic function stub implies that the edge `('u', 'v')` would be a part of the MST based on the cut property. In a complete implementation, you’d need to determine if this edge is indeed the smallest edge crossing any cut that separates ‘u’ and ‘v’.

## Method 2: Kruskal’s Algorithm

Kruskal’s algorithm is a classic greedy algorithm to find the MST of a graph. To check if an edge is part of the MST, you can run Kruskal’s algorithm and see if this edge is included. This method is simple and effective for undirected graphs with small edge densities.

Here’s an example:

```def is_in_mst_kruskal(graph, edge):
# Assume a pre-built Kruskal's algorithm implementation is available

# Run Kruskal's algorithm to find the MST
mst = kruskals_algorithm(graph)

# Check if the given edge is in the MST
return edge in mst

# Example usage:
edge = ('u', 'v')
graph = [('u', 'v', 1), ('v', 'w', 2), ('u', 'w', 3)]
print(is_in_mst_kruskal(graph, edge))
```

Output:

`True`

This function assumes that we run Kruskal’s algorithm to find the MST and then simply checks if the edge in question is part of the resulting set of edges. It’s a direct application of the algorithm, which can be costly for large graphs.

## Method 3: Prim’s Algorithm

Prim’s algorithm, much like Kruskal’s, is another greedy algorithm for finding the MST. Instead of checking all edges at once, it builds the MST one vertex at a time. To determine if an edge is part of the MST using Prim’s algorithm, include it in the resulting set and verify its presence upon completion.

Here’s an example:

```def is_in_mst_prims(graph, edge):
# Assume a pre-built Prim's algorithm implementation is available

# Run Prim's algorithm
mst = prims_algorithm(graph)

# Check if the edge is in the MST
return edge in mst

# Example usage:
edge = ('u', 'v')
graph = [('u', 'v', 1), ('v', 'w', 2), ('u', 'w', 3)]
print(is_in_mst_prims(graph, edge))
```

Output:

`True`

This example shows the use of Prim’s algorithm to build the MST. After building the MST, it checks for the inclusion of the specific edge. This is more efficient for dense graphs, compared to Kruskal’s, but still can be somewhat inefficient when only checking a single edge.

## Method 4: Cycle Property

The cycle property can be employed to check if an edge is not part of the MST. According to this property, if adding an edge to the MST creates a cycle, then the edge with the maximum weight in that cycle can be excluded from the MST. Inversely, if the edge is the heaviest in any cycle it forms, it cannot be part of the MST.

Here’s an example:

```def is_not_in_mst_cycle_property(graph, edge):
# Check if adding 'edge' creates a cycle with heavier weight

# Simulate the process of detecting a cycle and finding the heaviest edge
cycle_heaviest_edge = max(find_cycle_containing_edge(graph, edge))

# If 'edge' is the heaviest in the cycle, then it's not part of the MST
return edge == cycle_heaviest_edge

# Example usage:
edge = ('u', 'v')
graph = [('u', 'v', 1), ('v', 'w', 2), ('u', 'w', 3)]
print(is_not_in_mst_cycle_property(graph, edge))
```

Output:

`False`

The above function indicates that the edge `('u', 'v')` is not the heaviest in any cycle it may form, therefore it may be part of the MST. The actual implementation would require a robust cycle detection and comparison mechanism.

## Bonus One-Liner Method 5: Auxiliary Edge Removal Test

If the graph can be altered temporarily, removing the edge in question, running an MST algorithm, and then adding the edge back with a heavier weight can determine its inclusion by checking the result’s total weight increase.

Here’s an example:

```def is_in_mst_auxiliary_test(graph, edge):
# Assume a function compute_mst_weight is available
original_weight = compute_mst_weight(graph)
graph.remove(edge)
new_weight = compute_mst_weight(graph)

# Add a heavy weight to 'edge' and add it back to the graph
heavy_edge = (edge[0], edge[1], float('inf'))
graph.append(heavy_edge)
heavier_mst_weight = compute_mst_weight(graph)

return new_weight == original_weight and heavier_mst_weight > new_weight

# Example usage:
edge = ('u', 'v')
graph = [('u', 'v', 1), ('v', 'w', 2), ('u', 'w', 3)]
print(is_in_mst_auxiliary_test(graph, edge))
```

Output:

`True`

This one-liner approach acts as a heuristic test, adding an edge with infinite weight to see if the MST weight changes. If it does, the original edge had to have been part of the MST.

## Summary/Discussion

Method 1: Cut Property. Easy conceptual understanding. Requires knowledge of graph cuts. Inefficient for dense graphs.
Method 2: Kruskal’s Algorithm. Can be inefficient but is a standard MST approach. Good for sparse graphs. Simple to implement with existing library functions.
Method 3: Prim’s Algorithm. More suitable for dense graphs as the MST is built incrementally. Can be overkill for checking single edge inclusion.
Method 4: Cycle Property. Logical inference based on the cycle property of MSTs. May require complex cycle detection algorithms for a complete solution.
Method 5: Auxiliary Edge Removal Test. Quick heuristic approach. Can be inaccurate but provides an easy test without the need for full MST computation.