Questions tagged [spanning-trees]
The spanning tree of a connected undirected graph G is a tree having all the vertices and some number of edges of G.
127
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3answers
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Algorithm to get any spanning tree not necessarily a minimum spanning tree
Is there an algorithm to find a spanning tree. I know that there are $n^{n-2}$ of them and we have algorithms to find a minimum spanning tree.
But what if I just want any spanning tree? It doesn't ...
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2answers
53 views
Is the inverse of MST cut property true? Why?
If we partition the nodes of a graph into sets A and B, there is an edge e of weight larger than any other edge crossing the cut between A and B, e would never be in the minimum spanning tree?
0
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1answer
38 views
Unsure why (or whether?) a certain algorithm correctly computes a Minimum spanning tree
CLRS problem 23-4 part c gives an algorithm that may or may not compute a minimum spanning tree. Given some connected undirected graph G, we have
...
1
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1answer
40 views
Finding an algorithm that minimizes vertex weight sum of a subgraph that satisfies several constraints
I have a vertex-weighted undirected graph $(V,E)$ with root vertices $R = {r1, ..., rn}$. I need to find the subset $V'⊂V$ such that $R⊂V'$, $N[V']=V$, $∀v'∈V '[∃r∈R ($path($r', v'$)$)]$ that ...
0
votes
1answer
49 views
Undirected graph whose BFS and DFS trees have roots of degree 2
Draw a graph on $5$ vertices that satisfies all of the following conditions:
$G$ is an undirected connected graph.
For every node $v∈V$, in the spanning tree received by BFS($v$), $\deg v=2$.
For ...
0
votes
1answer
35 views
Number of spanning arborescences with a specific root in a directed graph
I am wondering how to calculate the number of spanning arborescences in a directed graph when a root is specified. For example:
where there are 5 spanning arborescences. Note that there is an edge ...
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0answers
18 views
Maximizing a spanning tree in an undirected graph with double weighted edges
I would like to have help in developing the algorithm for this problem.
0
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1answer
78 views
Finding minimum possible cost of road network between cities with distance from capital condition
I have a graph G containing cities (vertices V) connected by distanced roads (weighted undirected edges E).
Characteristics of the graph:
Each city is connected to the rest of the graph
Each city ...
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votes
1answer
181 views
maximum spanning tree in a complete graph
Given a complete graph how do I find maximum weight spanning tree.
where $weight(u, v) = \sum_{i=1}^{k} |w_{i,u} - w_{i,v}|$
assuming $k \lt 7$ and $n \le 500000$.
$n$ number of nodes
$weight(u,v)$ ...
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1answer
40 views
Prove following statement about Kruskal Algorithm
Let G be undirected graph, G=(V,E), and all edge weights are distinct. Consider an edge e=(u,v)∈E that wasn't included in the solution obtained from applying Kruskal Algorithm to G. Prove that this ...
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0answers
47 views
Total weight of all spanning trees
Given a weighted simple undirected connected graph $G = (V, E, w:E \to \mathbb{R})$, let $\tau(G)$ be the set of all its spanning trees. Is there an efficient algorithm to determine or estimate with ...
1
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1answer
148 views
Minimum spanning tree of multi directed graph
I have problem of inferring a rooted tree out of a connected simple graph.
The inference can be done by finding its minimum spanning tree, but the result is restricted by additional two types of ...
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0answers
19 views
Mathematics of the upper bound for the encoded input of an instance Kruskal's MWST problem
I was going through the classic text "Introduction to Automata Theory, Languages, and Computation" by Hofcroft, Ullman and Motwani where I came across the encoding of an instance of Kruskal's Mininum ...
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3answers
45 views
Variation to spanning tree called Group Spanning Tree
Suppose we have a complete graph, with say 100 nodes. We divide the nodes in the graph into groups, for example 10 nodes in each group, identified by color. We want to obtain a minimum spanning tree ...
4
votes
1answer
53 views
Count bridging edges in a family of two component forests
I am given a (simple, undirected, connected) graph $G = (V, E)$ and a fixed spanning tree $T$ in this graph. Removing an edge $e\in E(T)$ from $T$ splits it into a spanning forest $F^e$ with two ...
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0answers
127 views
MST with possibly minimal diameter
I am working with some research problem connected loosely to TSP which requires to find the Minimum Spanning Tree of a fully connected, weighted graph, where all the weights are positive and the graph ...
1
vote
2answers
704 views
Graph with exactly 2 Minimum Spanning Trees
Say that a graph, $G = (V, E)$ has 2 minimum spanning trees (MSTs).
Given this condition stipulated, prove that any cycle formed by all
the edges in both the MSTs (i.e., the union of the edges in ...
3
votes
1answer
71 views
how to generate all spanning trees from one spanning tree
If I have one spanning tree from a connected and undirected graph, how can I generate all other spanning trees of this graph by modifying this spanning tree one edge at a time? All intermediates must ...
2
votes
1answer
221 views
Kirchhoff's Spanning Tree Algorithm
Recently I have studied Kirchhoff's spanning tree algorithm to count the number of spanning trees of a graph, which has the following steps:
Build an adjacency matrix
Replace the diagonal entries ...
0
votes
1answer
23 views
Finding most likely tree over a semilattice
If I am not mistaken, then a semilattice defines a finite set of trees, for example spanning trees.
Now assume that each semilattice edge is annotated with a transition probability. In addition, let'...
2
votes
1answer
38 views
Number of spanning trees in undirected simple graph
What is the number of spanning trees in an undirected simple graph?
My attempt:
Let $m$ be the number of edges in a simple graph, and let $n$ be the number of vertices.
Then number of spanning ...
0
votes
1answer
33 views
Name for Turning DAG into redundant tree
I am looking for a term:
How is the tree called that you can obtain from a DAG by going top-down and appending all visited nodes to a tree, thereby copying nodes from the DAG into multiple occurences ...
1
vote
1answer
107 views
Average branching factor of an undirected graph
I'm trying to determine, given an unweighted undirected graph, the maximum number of leaves of any travelling of the graph, which means, the maximum number of leaves among all traversals of every ...
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2answers
54 views
First time visited nodes form a spanning tree that has a same number of edges in both BFS and DFS
I am trying to state, whether the statement is true: During a DFS/BFS, first time visited nodes form a spanning tree, that has the same number of edges whether you use DFS or BFS. Is it true?
What I ...
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1answer
253 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 ...
2
votes
1answer
946 views
Is maximum-leaves spanning tree np-complete?
How can we show that a maximum-leaves spanning tree is NP-complete? what other np-complete problem we can use as our reduction base?
(maximum-leaves spanning tree: does G have a spanning tree with at ...
0
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0answers
17 views
Intuitively, my problem is a mix of perfect hashing, tree spanning, combinatorial stuff - Ordered Decision Tree?
The problem I'm trying to solve is difficult to to give a single name, but I'll call it the ordered decision tree problem.
Imagine a row of commands:
...
0
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1answer
66 views
The role of the root switch after Spanning Tree Protocol has established a tree network in a LAN?
In Spanning Tree Protocol, a root switch is selected at first, and then somehow, the shortest path from each other switch to the root is obtained. Thus we established a tree network.
My questions ...
0
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2answers
605 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 ...
2
votes
1answer
4k views
DFS & BFS Spanning Trees
I want to construct a DFS and a BFS spanning trees for the graph below. The root is node a. At each step the next edge to be traversed should be the cheapest one.
DFS:
My understanding that to the ...
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votes
7answers
4k views
Do any two spanning trees of a simple graph always have some common edges?
I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to ...
2
votes
2answers
424 views
Christofides algorithm (by hand) (suboptimal solution - is it my fault?)
I would like to calculate an eularian path using Christofides algorithm on this graph: (Focus on the first number in each box representing the distance)
$\alpha$ denotes the start and end vertex of ...
0
votes
1answer
54 views
Proving equivalent definitions for MSTs
I am working on the following homework exercise:
Let $G = (V,E)$ be an undirected graph and $c: E \rightarrow \mathbb{R}$ it's cost function. Further let $T = (V,E')$ be a spanning tree in G.
I need ...
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0answers
77 views
Form a Tree having minimum diameter
I am given a connected graph. I have to construct a spanning tree from the graph, that has minimum diameter.
However, I looked for the solution, and the solution goes like this.
If the diameter of ...
2
votes
1answer
74 views
How to find MST for each source
Let's say I have a map with factories and selling points. I want to trace the paths from factories to the selling points with the lower possible cost.
The image bellow is an example of a possible ...
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vote
0answers
106 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, ...
1
vote
0answers
65 views
Spanning tree with equally separated edge weights
I have a fully-connected graph $G=(V,E)$ with edge weights $w(v)\in\mathbb{R};v\in V$ and I need to find a spanning tree $T=(V_t\subseteq V,E_t\subseteq E)$ where the set of edge weights in the tree ...
1
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1answer
64 views
Transforming undirected maximum spanning tree into directed augmented network
I am having trouble transforming a maximum weighted spanning tree into a directed tree such that each node is allowed at most one parent node. Taken from page 141 Friedman et. al (1997), the outline ...
3
votes
1answer
115 views
Survival algorithm for Network deterministic failures
Consider an undirected network $G = (V,E)$ in which edge $e$ $\in$ $E$ fails after (deterministic) time $t(e) > 0$. Network failure occurs at the first instant in which $G$ is no longer connected. ...
1
vote
1answer
182 views
Minimum sub-tree of a graph that covers each color at least once
I have a connected graph $G$ with $k$ different colors assigned to $n$ nodes where $k<n$. All edges have unit weight.
I want to figure out an algorithm to find a minimum sub-tree of $G$ that ...
3
votes
2answers
362 views
Is every edge of a graph included in some spanning tree?
Let's say we have a graph $G$. We pick one edge from it (any edge). Will there always be such a spanning tree that contains that very edge?
I think the answer is yes, because no matter what we do we ...
4
votes
3answers
830 views
Algorithms for procedural generated mazes
For the purposes of this question, a maze is a spanning tree on a square grid (although the type of grid isn't super important).
There are many Maze generation algorithms, but they only work on a ...
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0answers
39 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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1answer
1k views
Number of rooted labelled trees
According to Cayley's formula, we have number of spanning trees on a complete graph as n^n-2 and number of labelled trees with n vertices as n^n-2
If the tree is rooted then in each tree we can ...
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2answers
8k views
Minimum spanning tree using DFS and BFS
Can we construct minimum spanning tree for an undirected graph with distinct weights using bfs or dfs?
I have gone through many answers but each answer says something different and I am not convinced....
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votes
1answer
107 views
Min spanning tree that preserves total weight of original graph
I have a directed, weighted graph with no double edges. Each node represents a person, and each edge represents a debt. I want to reduce the total number of transactions required to settle all debt, i....
2
votes
1answer
4k views
safe edge for Minimum spanning tree
I know this question is posted many times, but i am still posting this because i have my own doubt which pulls me out to move forward to kruskal and prim's algorithm.So Please do help me out $-:$
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3
votes
1answer
242 views
Computational complexity of finding a spanning tree in planar hybergraphs
Using a bipartite graph to represent hypergraphs as described by Wikipedia :
A hypergraph $H$ may be represented by a bipartite graph $BG$ as follows:
the sets $X$ and $E$ are the partitions of $...
0
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0answers
150 views
Spanning-Tree-Protocol and BFS? (Distributed Computing)
Given is graph with networks and bridges/switches. We know, the root
Bridge is the bridge with the minimal Bridge-ID. The connections between every bridge and network is 1.
In the lecture slides they ...
0
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0answers
141 views
Distributed MST in $O(n \log \log n)$
I'm facing the following problem:
Describe a distributed MST algorithm in time $O(n \log \log n)$
I've managed to think of the following,
Run GHS(Gallager, Humblet and Spira) algorithm, till there ...
