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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22
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3answers
18k views

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 ...
13
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1answer
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 ...
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2answers
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 ...
3
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1answer
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 ...
7
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2answers
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 ...
3
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1answer
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 ...
4
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3answers
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 $G$ (...
0
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1answer
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 ...
0
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1answer
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). ...
2
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1answer
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 ...
5
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2answers
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 ...
5
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2answers
1k views

Find all critical edges for minimum spanning tree

This is a problem from the textbook "Algorithms, 4th edition" by Robert Sedgewick and Kevin Wayne. 4.3.26 Critical edges. An MST edge whose deletion from the graph would cause the MST weight to ...
2
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1answer
409 views

When is the output of shortest path $\subset$ MST?

I was wondering if the output of an algorithm like Dijkstra was always contained in the minimal spanning tree, however, a counter example to this claim are cyclic graphs like: The shortest path $B \...
1
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1answer
660 views

Christofides algorithm: why must an MST have even number of odd-degree vertices?

This question is not necessarily related to Christofides algorithm per se, I just ran into it when reading about it. I assume that a minimum spanning tree must have an even number of odd-degree ...
1
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1answer
2k views

What edges are not in any MST?

This is a homework question. I do not want the solution - I'm offering the solution I've been thinking of and wish to know whether is it good or why is it flawed. Consider a weighted undirected graph....
4
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1answer
282 views

Are all minimum spanning trees optimized for fairness?

I know by definition that a minimum spanning tree (MST) of a weighted, connected graph has the lowest global value for for sum of all edges in a path that connects all vertices. I'm curious if all ...
4
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2answers
629 views

Diameter-constrained Minimum Spanning Tree Problem

The diameter-constrained Minimum Spanning Tree (MST) problem is as follows: you have a undirected weighted graph $G = (V,E)$ of different weights where $V$ is the set of vertices and $E$ is the set of ...
4
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1answer
442 views

Minimum spanning tree formulation as integer program

The minimum spanning tree problem can be solved in polynomial time via Kruskal's or Prim's algorithm. However, every integer program I have seen that corresponds to the MST problem require a ...
4
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1answer
334 views

Alternative algorithm for minimum spanning tree construction

Let $\textit{G(V,E)}$ be an undirected connected graph with distinct costs on its edges. Initialize $\textit{T}$ to be any spanning tree of $\textit{G}$. Consider an algorithm which replaces an ...
2
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3answers
188 views

Need Help Understanding MST Cutset Formulation

I just started learning about linear programming in my class, and I'm having some trouble understanding the MST Formulation Integer Linear Programming (Cutset Formulation). This is the definition: ...
2
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1answer
151 views

Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?

I am trying to prove the algorithm for Question 5 in this practice exam. I am trying to prove this algorithm with the following three claims: Suppose we have a graph G, a minimum spanning tree T, ...
1
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1answer
528 views

Is this “cut condition" sufficient for unique minimum spanning tree?

Given a connected, undirected, weighted graph $G$, the following "cut condition" For any partition of the vertices of $G$ into two subsets, the minimum-weight edge with one endpoint in each subset ...
1
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2answers
68 views

Minimum spanning tree such that one edge can be minimised

During a computer coding exam, I have encountered such a problem. Given a list of vertexes and edges between the vertexes,and a positive number, D, what is the minimum spanning tree between the ...
0
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1answer
3k views

Kruskal's and Prim's algorithm [duplicate]

Does Kruskal's and Prim's algorithm work on directed graphs? I want the two algorithms to find a minimum spanning tree. For further enlightenment, I would like to know what other problems Kruskal's ...
0
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1answer
355 views

Turn MST of G to MST of G with one new edge

Given $T$, an MST of $G(V,E)$ connected and undirected. Assume we add an edge $e'$ with weight $w(e')$. Suggest an algorithm which takes $T$ as input, and outs $T'$ MST of $G'(V,E\cup\{e'\})$.So i ...