# 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.

137 questions
Filter by
Sorted by
Tagged with
1 vote
20 views

### 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 ...
31 views

### 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 $2$, ...
1 vote
39 views

### 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) (...
• 165
565 views

### 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 ...
• 71
39 views

### 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 $U$ ...
28 views

### 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 ...
21 views

### 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 ...
38 views

### 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 $\operatorname{bwd}(P) = \min_{1 \le i \le k} f(e_i)$ ...
100 views

### 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 ...
• 115
34 views

### 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 ...
1 vote
36 views

### 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 ...
• 11
1k views

### 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 ...
1k views

### 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 ...
1 vote
110 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?
• 37
1 vote
120 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 ...
• 818
1 vote
108 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 ...
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 ...
• 145
60 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 ...
• 121
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 ...
223 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 $weight(u,v)$ ...
81 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 ...
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 ...
• 141
1 vote
366 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 ...
• 121
52 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 ...
59 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 ...
• 141
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 ...
1 vote
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 ...
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 ...
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 ...
• 1,162
24 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'...
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 ...
38 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
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 ...
• 415
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 ...
• 111
1 vote