Question

Show that a tree has either one or two centers.

   Show that a tree has either one or two centers.

Discrete Mathematics

Discrete Mathematics

Richard Johnsonbaugh 8th Edition

Chapter 9, Problem 36

Instant Answer

Solved by Expert Prashant Bana

03/17/2021

Step 1

First, we need to understand what a center of a tree is. A center of a tree is a node that, if removed along with its incident edges, would result in the minimum possible maximum distance from any remaining node to the remaining connected component. Show more…

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Show that a tree has either one or two centers.

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Key Concepts

Tree (Graph Theory)

A tree is a connected graph with no cycles, meaning that there is exactly one unique path between any two vertices. This fundamental structure in graph theory is widely used due to its simplicity and properties, which often simplify problems like finding centers or optimal paths.

Graph Center

The center of a graph is the vertex (or vertices) with the minimum eccentricity, where eccentricity is defined as the greatest distance from a vertex to any other vertex in the graph. In trees, this concept leads to particular structural properties that allow for a clear determination of the center.

Leaf Removal Process

A common technique to identify the center of a tree involves iteratively removing leaf vertices, which are vertices of degree one, until only one or two vertices remain. This process leverages the tree's hierarchical structure to gradually peel away the outer layers, revealing the central point(s) of the tree.

Uniqueness of Paths in Trees

Since trees have the property that there is a unique path between any pair of vertices, measuring distances and hence eccentricities becomes straightforward. This uniqueness is critical in proving that the process of removing leaves will always result in either one unique center or two adjacent centers, as there are no alternate paths that could complicate the distance measurements.

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