# Tagged Questions

For questions about trees in graph theory, which are connected graphs with no cycles. Also can be used for questions about forests, which are graphs that are disjoint unions of trees.

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### Bottleneck distances for the Steiner problem in graphs

I have been reading the paper "Preprocessing the Steiner Problem in Graphs" by Duin (http://link.springer.com/chapter/10.1007%2F978-1-4757-3171-2_10) and I am having a bit of trouble wrapping my head ...
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### Tree decomposition (Citation needed)

Recently, I read the statement "Fix $k\geq 1$, Any tree with at least $k$ edges may be decomposed as a union of edge-disjoint subtrees, each having between $k$ and $3k$ edges" Now I was wondering ...
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### Number of reachable vertices in a tree

Given a tree $T$ with infinite nodes. Each node of the tree has exactly $C$ children. I need to figure out that, starting from a node at distance $h$ from root, how many distinct vertices can be ...
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### Find all possible topological-sortings of graph G

A topological ordering of G is an ordering of the nodes as $v_1,v_2,...,v_n$ so that all edges point "forward": for every edge $(v_i,v_j)$, we have $i<j$. Moreover, the first node in a ...
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### Trees with no vertex of degree 2 have more leaves than internal nodes

There is a question asked by portal about Tree having no vertex of degree 2 has more leaves than internal nodes so we want to prove this claim by induction and an answer from Micheal Biro suggested ...
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### What is a “linear chain” in Graph Theory?

What is a linear chain in the context of graphs and trees? For example: a topological sort forms a linear chain What does a linear chain mean in the example above? Another example from ...
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### How can I draw a tree to represent combinations?

I understand how to systematically draw a tree for permutations. How do you do this for combinations? In my book, I don't see a system to avoid repetitions. I'd like to draw a tree of 5C3 if possible. ...
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### Circuits and Trees

Given a graph G, can it be split into 2 sets of graphs($G_1, \; G_2$) such that, $G_1$ consists only trees and $G_2$ consists only circuits ? In other words: Is it possible to construct any graph ...
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### Tree having no vertex of degree 2 has more leaves than internal nodes

If $T$ is a tree having no vertex of degree 2, then $T$ has more leaves than internal nodes. Prove this claim by a) induction, b) by considering the average degree and using the handshaking lemma. I ...
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### There are at least 22 vertex-disjoint paths between every pair of vertices?

$G$ is a graph on $n$ vertices and $2n−2$ edges$.$ The edges of G can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for $G?$ For every subset of $k$ ...
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### Intersection of all possible spanning trees of a connected, simple graph.

Is the intersection of all possible spanning trees of a simple, connected graph $G$ equal to the graph $(V_{G}, \varnothing)$? I'm not sure if this is a trivial question or not. Although I'm going to ...
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### Why is the intersection of countably many homogeneously Suslin subsets of $\,^{\omega} \omega$ homogeneously Suslin?

I found this assertion in these notes: The derived model theorem (Steel) right in the beginning on page 3, together with the remark that this is 'not too hard to show'. Unfortunately, I'm ...
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### Tree with no nodes of degree 2: prove that # leaves ># internal nodes using average degree and handshake lemma

Im really struggling to formalise my thoughts on this one. Basically I understand that if we would allow nodes with degree 2, then we could chain together infinitely many nodes to always produce ...
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### Different trees of weighted graph , please check whether my explanation is correct $?$

Let G=(V, E) be a graph. Define $\xi(G) = \sum\limits_d i_d*d$, where $i_d$ is the number of vertices of degree $d$ in G. If S and T are two different trees with $\xi(S) = \xi(T)$, then ...
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### Complexity analysis of alpha beta pruning of a full tree

I am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...
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### Yet another curious convolution

Some time ago, I found the following algorithmic problema: Count the number of distinct unrooted, unordered, labeled trees of $n$ nodes where each node has at most $k$ neighbors. Given that the ...
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### For a k-ary tree with height h, the upper bound for the maximum number of leaves is $k^h$

i want to prove For a k-ary tree with height h, the upper bound for the maximum number of leaves is $k^h$. (assume the complete k-ary tree is a tree that is complete in all levels including the ...
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### directed trees versus singly-connected directed graphs

According to wikipedia, a singly connected directed graph (a.k.a. Polytree) is a Directed Acyclic Graph whose underling undirected graph is a tree. How is this different from directed trees (trees ...
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### Is there a name for this particular kind of tree graph?

I've recently encountered a problem which heavily involves analysis of structures analogous to weighted trees with no nodes of degree two (such a node along with its adjacent edges would be ...
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### Binary Minimum Spanning Tree (from complete graph)

Given a weighted complete graph (or more exactly, a matrix of pairwise metric distances between vertices), I need to find a good approximation of the binary spanning tree of lowest total cost. There ...
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### Graph Theory: Trees, leaves and cycles

So, a vertex is called a leaf if it connected to only one edge. a) Show that a tree with at least one edge has at least 2 leaves. b) Assume that G = (V, E) is a graph, V ≠ Ø, where every vertex ...
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### Counting spanning trees and hamiltonian paths

Assumption : For any connected graph, every hamiltonian path is a spanning tree but not the other way around. If the assumption is wrong there is no need of reading any furhter. So is the assumption ...
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### Extract the overall “structure” (“backbone”) of a set of vertices…

My main problem is that I am struggling to find a good graph-theoretical formulation of my problem, let alone a formal name for it (if it already exists, as I suspect it should)… Informally, given a ...
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### Name of operation: changing the root of a rooted tree

What is (if any) the name of the operation of changing the root of a rooted tree? Picking a vertex which is not the root, then reorienting the edges in such a way that the vertex becomes the root?
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### is the Root of a binary Tree counted as a node

I am working on this Homework questions and there's one thing I can't seem to understand. We are trying to proof using structural induction that some elements in T hold for the following statement :...
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Problem Consider the general chip-and-be-conquered recurrence relation: $T(n) = b_1T(n - 1) + b_2T(n - 2) + ... + b_kT(n - k) + f(n)$; for $n >= k$ for some constant $k >= 2$. The ...
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### How would one go about solving these types of problems?

I'm totally lost. All I know is it has to do with binary trees and may need to be solved using induction. Show that every 2-tree with $n$ internal nodes has $n + 1$ external nodes. Show that the ...
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### Terminology for excluded nodes

Given the following tree: A / \ / \ B C / \ \ / \ \ D E F / \ / \ G H Regarding node ...