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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 ...
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### 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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### How to find spanning tree of a graph that minimizes the maximum edge weight?

Suppose we have a graph G. How can we find a spanning tree that minimizes the maximum weight of all the edges in the tree? I am convinced that by simply finding an MST of G would suffice, but I am ...
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### Find a MST such that it's mostly red (original graph's edges are colored red and blue)

Consider the following problem: Given a simple, strongly-connected, weighted graph G=(V,E), of which every edge is colored either red or blue (in addition to having a numeric weight). Find an ...
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### 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 ...
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### 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 $G$ (...
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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 $B \... • 1,483 1 vote 1 answer 682 views ### Do the minimum spanning trees of a graph have the same number of edges with a given weight? I'm asking about the answer here: Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight? I didn't understand the best answer here Choose edge$e \in ...
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For my homework I have a problem that I can't solve and it makes me wonder about 2 different MST: Let $G=(V,E)$ be a graph that has a minimum spanning tree $T$. I want to find another minimum ...
### Why if $G$ has two spanning trees $A$ and $A'$, then every edge of $A'\cup \{e_i\}\in A'$
Theorem: Let be $G$ a weighted graph in which every edge has a different weight. Suppose that $G$ has two spanning trees $A$ and $A'$. Let be $i$ the first index such that $e_i\ne e'_i$ ...