# Questions tagged [minimum-spanning-tree]

Use this tag whenever your question is related to minimum spanning tree (MST). An MST of a connected edge-weighted graph G is a spanning tree whose sum of edge weights is as small as possible. We usually assume $G$ is finite, simple and undirected.

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### When is the minimum spanning tree for a graph not unique

Given a weighted, undirected graph G: Which conditions must hold true so that there are multiple minimum spanning trees for G? I know that the MST is unique when all of the weights are distinct, but ...
2k views

### Are all MST minimum spanning trees reachable by Kruskal and Prim?

I believe this is true but have not been able to get a formal proof for either. But is it true that any minimum spanning tree is reachable by applying Kruskal's algorithm? Similarly, is this true for ...
266 views

### Equivalent definition of minimal spanning tree

Prove that $T$ is MST $\Leftrightarrow$ for any edge $uv \notin T$, $uv$ has the maximal weight on the cycle created by adding $uv$ to $T$. It's my attempt to prove $\Rightarrow$: Consider the cycle ...
5k views

### Why do we have different algorithm for MST when graphs are directed?

What was the reason to come up with Chu–Liu/Edmonds' algorithm when the input graph is directed instead of using the Prim's or Krushkal's method for finding Minimum spanning tree ? What cases are not ...
5k views

### Updating an MST $T$ when the weight of an edge not in $T$ is decreased

Given an undirected, connected, weighted graph $G = (V,E,w)$ where $w$ is the weight function $w: E \to \mathbb{R}$ and a minimum spanning tree (MST) $T$ of $G$. Now we decrease the weight by $k$ of ...
142 views

### Between every two MST's there's a series of “nearby” MST's

Given undirected connected graph $G=(V,E)$ and a weight function $w:E\to\mathbb{R}$, two MST's $T_1, T_2$ are nearby if there exists $e\in T, e'\in T'$ such that $T'=(T-\{e\})\cup\{e'\}$. Prove that ...
420 views

### Are edges in a minimum spanning tree not heavier than respective edges in another spanning tree?

Let $G$ be an undirected connected weighted graph, and let $T$ be a minimum spanning tree of $G$ with edge weights: $w_1 \le w_2 \le ... \le w_{n-1}$. Now let $T'$ be some other spanning tree of $G$ (...
4k views

### MST: Are all safe edges, light edges?

Following are some definitions from CLRS: DEFINITIONS : 1. Cut (S ,V-S) : of an undirected graph G = (V,E) is a partition of V(as defined in CLRS Book) .You can think it as a line that divides ...
380 views

### Is this “cycle” condition sufficient for unique minimum spanning tree?

Given a connected, undirected, weighted graph $G$, the condition The maximum-weight edge in any cycle of $G$ is unique. is not necessary for $G$ to have a unique minimum spanning tree (MST). ...
618 views

### Prim's algorithm: difference between brute force and PQ approaches

I'm trying to figure out the different way we obtain an MST with a brute force Prim's algorithm compared to the optimized version based on priority queues. Given a graph $G=(V,E)$, the former can be ...
581 views

### MST with half the edges the maximum weight

I have been cracking my head over the following question - You are given an undirected connected graph with an even number of edges. Half of the edges have weight less than C (possibly with ...