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.

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safe edge theorem proof clarification

I found the following proof for the theorem that states "A light edge that crosses a cut that respects A is safe for A": See: https://www2.hawaii.edu/~janst/311_f19/Notes/Topic-17.html ...
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Recursively deleting spanning forest from graph, how many iterations maximum to get to the empty graph?

As in the question stated, I am interested in the approximation factor of the greedy approach to compute the arboricity of the graph. My intuition tells me the factor should not be bigger than $2$, ...
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Recreate a spanning tree in a grid graph given vertex descriptions

Let's assume I have graph above with spanning tree pointed out by blue edges. Vertex at position (1,1) (row 1, column 1) is connected to the bottom vertex and has degree 1. Vertex at position (4,2) (...
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Spanning tree - minimum difference between smallest and largest weight

I am given an undirected, weighted graph $G$, on its base I have to create a spanning tree with such a property that the difference between the largest edge weight and the smallest edge weight is the ...
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Wrong Solution for `Spanning tree with chosen leaves` problem

Suppose that we're given a connected, undirected graph $G = (V, E)$ with edge weights $w_e$ and a subset of vertices $U \subset V$. We want to find the lightest spanning tree in which the nodes of $U$ ...
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The Roskind-Tarjan Algorithm

I am going through the paper https://pubsonline.informs.org/doi/abs/10.1287/moor.10.4.701 which is A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees and the authors are James Roskind and ...
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Edmond's theorem for k-disjoint arborescences in digraphs

Recently while studying arborescences in graph theory, I came across Edmond's theorem for $k$ edge-disjoint arborescences in digraphs if a finite digraph is $k$ edge-connected from a vertex r for ...
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Spanning tree that maximizes all-pairs bandwidth => Maximum spanning tree?

Let $G = (V, E)$ be a weighted, undirected graph, with $f: E \to \mathbb{R}$ its weight function. Given a path $P = (e_1, \dots, e_k)$, we call $\operatorname{bwd}(P) = \min_{1 \le i \le k} f(e_i)$ ...
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Prove finding a spanning tree with no more than 50 leaves is NP-hard

This is a homework question. Consider the problem of finding if an undirected graph $G$ can have a spanning tree with no more than 50 leaves. Is this problem NP-hard? I think it is and I'm trying to ...
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Understanding the Rectilinear Spanning Tree Algorithm

I am required to find a Rectilinear Spanning Tree in $\mathcal{O(n\log(n))}$, where $n$ is the number of vertices to connect. A Rectilinear Spanning Tree is a spanning tree made of nodes where each ...
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Linear deterministic algorithm for finding spanning tree T with minimal maximum edge

Given an undirected connected graph $G = (V, E)$ with weights $w : $E → $R$$^+$, define for a spanning tree T the value $λ$(T) = $max_e$∈$T${w(e)} (the maximal edge weight in T ). I need to find a ...
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How does Dijkstra's problem 1 (tree of minimal total length) work and what does it do?

In Dijkstra's original paper, he talks about two problems related to graphs. The second one is the problem of finding the shortest path between two nodes, which is what is most commonly meant by ...
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2 votes
3 answers
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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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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?
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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 ...
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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 ...
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114 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 ...
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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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1 answer
247 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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1 answer
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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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1 answer
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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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2 votes
0 answers
54 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 ...
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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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3 answers
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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 ...
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4 votes
1 answer
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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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8 votes
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160 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 ...
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1 vote
2 answers
1k 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 ...
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3 votes
1 answer
102 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 ...
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2 votes
1 answer
381 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 ...
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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'...
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2 votes
1 answer
89 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 ...
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1 answer
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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 ...
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1 vote
1 answer
203 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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2 answers
108 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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1 answer
478 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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2 votes
1 answer
2k 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 ...
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0 answers
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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: ...
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1 vote
1 answer
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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 ...
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0 votes
2 answers
842 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 ...
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2 votes
1 answer
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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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25 votes
7 answers
5k 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 ...
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2 votes
2 answers
579 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 ...
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1 answer
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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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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 ...
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2 votes
1 answer
100 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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1 vote
0 answers
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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, ...
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1 vote
0 answers
67 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 ...
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  • 141
1 vote
1 answer
71 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 ...
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4 votes
1 answer
123 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. ...
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2 votes
1 answer
310 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 ...
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