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.
Questions tagged [minimum-spanning-tree]
254 questions
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Proof that "local" minimum spanning tree is "global" minimum spanning tree
I'm trying to understand a proof from the book "Graph Theory with Application to Engineering & Computer Science" by Narsingh Deo.
The chapter is about trees in non oriented graphs.
A bit of terminology so that you can understand the theorem and the…
DisplayName80
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If an edge e doesn't belong to any mst, then what can we say about e?
I'm trying to prove a statement "given a graph G=(V,T) and that no cycle C exists that contains only edge "e" and other edges that their weight is smaller than that of "e", prove that "e" must be in some mst".
I tried to proof by contradiction, but…
CSstudent
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MST with degree constraints on some node
While preparing for an algorithms exam I came across the following problem in a practice test:
Let $G = (V,E)$ be a connected, undirected graph with weighted edges (all weights are rational numbers but not necessarily all unique) and let $v$ be some…
Or Bairey-Sehayek
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Minimum spanning subgraph of even degree
I have an unusual problem that I am struggling to solve. I have a set of nodes (with positive distances between them) that I want to connect into a single component. In particular, I want to form a minimum-distance spanning subgraph such that all…
Dmitry Kamenetsky
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Is it possible for a maximum weight edge of a cycle being included in MST?
Let C be a cycle in a simple connected weighted undirected graph. Let "e" be an edge of maximum weight on C
Which of the following is TRUE?
(A) No minimum weight spanning tree contains e.
(B) There exists a minimum-weight spanning tree not…
Mr. Sigma.
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Minimum Spanning Tree with diameter 3
I need an $O(|V||E|)$ algorithm to detect if a graph has a minimum spanning tree with diameter 3.
This paper mentions that it is possible for any diameter by finding the vertex 1-center which can be done using Floyd's algorithm, but I need something…
Avi Tal
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Sensitivity analysis of $MST$ edges
I am working on the following exercise:
Consider an undirected graph $G = (V,E)$. Let $T^* = (V,E_{T^*})$ be a $MST$ and let $e$ be an edge in $E_{T^*}$. We define the set of all values that can be assigned to $w_e$ such that $T^*$ remains a MST as…
3nondatur
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Optimal Loading of a Hopping Airplane Minimum Cost Flow Problem
I have a Problem with an Optimal Loading a Hopping Airplane example . This is the Part of Minimum Cost Flow Problem.. .. I dont understand a Picture at all. I should to make one example with numbers and i need to show, how could i send flow along…
Dmitry Uvarov
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Why does the cut $(V_C,V-V_C)$ respect $A$?
Corollary 23.2
Let $G = (V,E)$ be a connected, undirected graph with a real-valued weight function $w$ defined on $E$. Let $A$ be a subset of $E$ that is included in some minimum spanning tree for $G$, and let $C = (V_C,E_C)$ be a connected…
Arka Prava Paul
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Whether the 2 minimum spanning tree of same graph contains the lowest edges in common?
If two minimum spanning trees on the same graph only have 2 edges in common, then those two edges must be the lowest costs edges in the graph.
True/false? and why?
Because according to me if there is a only edge which connects 2 subgraphs then…
AlgoMan
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Kruskal's and Prim's algorithm
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 and Prim's can solve. So far, I only know they can…
user26658
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Can the MST's "cut property" be derived from the "cycle property" and vice versa?
The cut and cycle properties of the minimum spanning tree are well-known. It is easy to use similar arguments to prove them. But I wonder if one property be derived from the other.
Cut property: Suppose edges X are part of an MST of G=(V, E). Pick…
Tom Bennett
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Edge being in the MST
Does an edge never being the largest weight in any cycle imply that it is included in the MST?
I believe the answer is not, but can't think of a counterexample. I know that one guarantees an edge is in the MST with the cut property, but was…
joeren1020
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Proving heaviest edge in mst is not heavier than heaviest edge of any possible spanning tree?
How to prove that the heaviest edge in mst is not heavier than heaviest edge of any possible spanning tree? by heaviest edge of any Spanning tree i mean considering any possible Spanning tree, we have to prove that the heaviest edge in mst is not…
vhd
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Prove a spanning tree M is an MST for G with weights c > 0 if and only if M is an MST with weights c^x for x > 0
Prove -
The spanning tree $M$ is a minimum spanning tree (MST) for $G$ with weights $c > 0$ if and only if $M$ is an MST for $G$ with weights $c^x$ for $x > 0$.
I am confused, how to prove this, as $c$ could be between $0$ and $1$, and then…
G K
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