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Chapter 9: Problem 36

Show that a tree has either one or two centers.

Short Answer

Expert verified
The proof involves applying the concept of the center of a tree and repeatedly removing leaves of the tree until no more can be removed. The outcome will always be either one or two vertices, which are the centers of the tree. Thus, it is proven that a tree can only have one or two centers.

Step by step solution

01

Understanding the centre of the tree

The center of a tree, T, can be defined as the minimum eccentricity of the vertices in T, where the eccentricity of a vertex, v, is the greatest distance from v to any other vertex. Remove the leaves (vertices connected by only one edge) of the trees and keep removing until only one or two vertices remains.
02

Keep removing the leaves

Consider any tree, T. We keep removing leaves until clear our tree. After the first removal of leaves, we're left with another tree (since it's still connected and has no cycles). This process can be repeated until we are left with either one vertex (if the original tree had an odd number of vertices) or two vertices (if the original tree had an even number of vertices).
03

Finalize the proof

Regardless of how we remove leaves, the remaining one or two vertices are those that minimize the maximum distance to every other vertex. It can be seen intuitively by noticing that they were 'protected' by the leaves that got removed in each step, showing that they are equally or less distant to each leaf. This means that in every tree, we will always end up with one or two vertices that are the center of the tree. Thus, we have now shown by this removing process, a tree has either one or two centers only. This finishes our proof.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tree Eccentricity
In graph theory, eccentricity is a crucial concept when discussing the structure of trees. A tree is a type of graph without cycles, and for a vertex, its eccentricity is defined as the maximum distance to any other vertex in the tree. Imagine you are at a particular point (vertex) in a forest (tree). The eccentricity would be the furthest you can walk to reach the farthest tree in the forest. It's like finding the boundary of your exploration from that starting point where you can still walk.

Calculating eccentricity for each vertex can help identify vital areas in the tree structure. This information is used to find the center of the tree, which is a vertex or collection of vertices minimizing the largest possible distance to other vertices. Hence, the center is not a mere geometric center but rather an optimal place relative to distances within the tree.

Understanding eccentricity helps in problem-solving contexts, such as optimizing communication in networks, where minimizing the maximum distance from vital nodes is crucial.
Leaf Removal Process
The Leaf Removal Process is an effective technique used to uncover the center of a tree. By methodically removing leaves鈥攙ertices connected by only one edge鈥攜ou simplify the structure and work towards identifying the center.

When you first remove the leaves, the tree transforms but retains its core interconnectedness and remains acyclic. By peeling away layer after layer of leaves, you gradually reduce the tree to its essential structure. Each removal step of leaves, you lessen the tree's number to eventually shrink to either one or two central vertices.

Why does the process stop at one or two vertices? This depends on whether the tree initially had an odd or even number of total vertices. After evenly stripping away the peripheral nodes, the center is naturally revealed as it is the last to be reached in terms of maximum distance. The leaf removal process reveals those vertices that remain consistently shielded by surrounding nodes until the very end.
Graph Theory Concepts
Graph Theory provides the foundational tools to understand complex networks through vertices and edges. In this context, a tree is a special type of graph that is acyclic and connected, making it a powerful tool for mathematical and real-world applications.

Essential to graph theory are concepts such as connectivity and acyclic nature, which help define trees. Graphs are visual structures that represent objects and their pairwise relations, allowing for the study of node connectivity and pathways.

Understanding how to determine the center of a tree using graph theory principles鈥攍ike eccentricity and leaf removal鈥攗nderscores the discipline's utility. These methodologies are essential not only in theoretical problems but also in practical applications like network design, where the efficiency of node communication is critical. For students of graph theory, mastering these concepts is a path to deeper insight into more complex graph structures.

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Most popular questions from this chapter

Refer to the following situation. Suppose that we have stamps of various denominations and that we want to choose the minimum number of stamps to make a given amount of postage. Consider a greedy algorithm that selects stamps by choosing as many of the largest denomination as possible, then as many of the second-largest denomination as possible, and so on. Find positive integers \(a_{1}\) and \(a_{2}\) such that \(a_{1}>2 a_{2}>1, a_{2}\) does not divide \(a_{1},\) and the algorithm, with available denominations \(1, a_{1}, a_{2},\) produces the fewest number of stamps to make any given amount of postage. Prove that your values do give an optimal solution.

Draw all nonisomorphic full binary trees having nine vertices.

Write a recursive algorithm that interchanges all left and right children of a binary tree.

Write an algorithm that returns the number of terminal nodes in a binary tree.

Let \(G=(V, E)\) be a simple undirected graph. \(A\) vertex cover of \(G\) is a subset \(V^{\prime}\) of \(V\) such that for each edge \((v, w) \in E,\) either \(v \in V^{\prime}\) or \(w \in V^{\prime} .\) The size of a vertex cover \(V^{\prime}\) is the number of vertices in \(V^{\prime} . A n\) optimal vertex cover is a vertex cover of minimum size. An edge disjoint set for \(G\) is a subset \(E^{\prime}\) of \(E\) such that for every pair of distinct edges \(e_{1}=\left(v_{1}, w_{1}\right)\) and \(e_{2}=\) \(\left(v_{2}, w_{2}\right)\) in \(E^{\prime},\) we have \(\left\\{v_{1}, w_{1}\right\\} \cap\left\\{v_{2}, w_{2}\right\\}=\varnothing\). Prove that for every \(n,\) there is a connected graph with \(n\) vertices that has a vertex cover of size 1.
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