# 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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### How do deal with the following situations using Prim's algorithm?

Consider the following Graph We want to generate the MST using Prim's algorithm. Starting from node A, suppose we pick B as our next node, we see a self-loop that has less weight than the two other ...
144 views

### Updating a mst after increasing the weight of an edge in the mst

Suppose we have a weighted undirected graph $G$ and a minimum spanning tree $T$ Let $G2$ be a new graph by increasing the weight of one edge $e = (a,b)$ that is part of $T$. I'm using a common ...
69 views

### Minimum spanning tree formulation

Can any expert explain the reasoning behind the constraint in the following formulation of the minimum spanning tree? To formulate the minimum-cost spanning tree (MST) problem as an LP, we ...
18 views

### Minimum Weight Binary Spanning Tree

Let $G=(V,E)$ be a simple graph with weights $w_{ij}$ (can be assumed to be positive). Is it possible to find the minimum (or maximum) weight, rooted spanning tree that is binary? That means every ...
787 views

### MST Of An Almost Tree

A graph $G = (V,E)$ is called an almost tree if it is connected and has most $n + c$ edges where $n = |V|$ and $c$ is a small constant number. How would I go about designing an algorithm for a given ...
101 views

### Build transportation system to travel between cities

Given $n$ cities, I'm looking to build a transportation system that allows travelling between every two cities. For every two cities $i$ and $j$, a road can be paved in the cost of $c_{ij}$. Also, ...
93 views

### n'th cheapest MST in a graph, with multiple edges that can have the same weight

I'm trying to think about an algorithm for this problem. I know there is an algorithm for the second cheapest MST in a graph, but if I understood it correctly it only solves cases in which every ...
40 views

### Is a spanning tree an MST if its weight can't be decreased by adding an edge and removing one? [duplicate]

My gut says it's true and I have tested it on a few examples. However, I can't prove it. I thought of using contradiction; suppose there exists another tree T' with smaller weight which has m edges ...
37 views

### Graph of MSTs of a graph - 2 msts connected if differ by 1 edge - is this single-component? [duplicate]

Suppose we take all MSTs of a graph and build a new graph where each vertex corresponds to a MST of the first graph and two vertices are connected if their corresponding spannings trees differ by only ...
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### All-pairs minimax path problem - MST [closed]

Let $G(V, E)$ be an undirected weighted (positive) graph. Given a path $s-t$ find the path that minimizes the maximum weight of any of its edges. This is the minimax path problem. It is know that a ...
281 views

### How can I modify Kruskal's algorithm to work with limited resources?

In Kruskal algorithm, I decided to use it to construct a minimum spanning tree for a set of N = 100,000 points. Since my hardware does not allow me to keep all the distances between pairs of points, I ...
80 views

### NP-hardness of Capacitated Minimum Spanning Tree and Price Collecting Steiner Tree on dag/tree

I am thinking about the NP-completeness of two graph problems on different graph structures. For example: The Capacitated Minimum Spanning Tree for graph is NP-hard. However, is the problem still ...
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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 ...
28 views

### Negative cycle detection using Bellman-Ford and its correctness

I recently started studying algorithms on my own using Cormen and MIT algo videos in YouTube. I am going thru Bellman-Ford. I have 2 doubts about the correctness of the algorithm: Why are we ...
478 views

### Minimum Distance Spanning Tree Dijkstra

I would like to construct a Minimum Distance Spanning Tree (Dijkstra) for the graph below: MDST: {(a,c), (c,h), (c,f), (a,d), (h,g), (a,b), (d,e), (h,j), (h,i), (j,k), (e,m), (i,l)} Is my ...