/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 The path connecting any two vert... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 23: Problem 14

The path connecting any two vertices \(u\) and \(v\) in a tree is unique.

Short Answer

Expert verified
The path between any two vertices in a tree is unique because assuming otherwise creates a cycle, which contradicts the tree definition.

Step by step solution

01

Understanding the Problem

We need to demonstrate that the path connecting any two vertices \(u\) and \(v\) in a tree is unique. A tree is a connected graph with no cycles.
02

Definition of a Tree

A tree is a connected acyclic graph, meaning there is exactly one path between any two of its vertices.
03

Suppose Two Paths

Assume, for contradiction, that there are two different paths between vertices \(u\) and \(v\) in a tree.
04

Contradiction via Cycle Formation

If these two paths are different, they must intersect at the endpoints \(u\) and \(v\) but diverge at some point to form a cycle, contradicting the definition of a tree which is acyclic.
05

Conclusion

Hence, assuming more than one path leads to a contradiction, proving that the path between any two vertices in a tree is indeed unique.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Tree Structure
A tree is a fascinating concept in graph theory with special properties. Unlike general graphs, a tree is characterized by having no cycles. This means if you were to walk along the nodes of a tree, you would never loop back to where you started.
One of the most striking features of a tree is its simple, hierarchical structure. You can envision it like an upside-down tree with branches cascading down from a single root.
Trees have important properties:
  • Each tree is a connected graph, meaning there are no isolated vertices.
  • They contain exactly one less edge than the number of vertices, i.e., if there are \( n \) vertices, then there are \( n-1 \) edges.
  • An important result of these properties is that there is precisely one path between any two nodes in a tree.
Connected Graph
A connected graph is a type of graph in which there is a path between every pair of vertices. This ensures that no part of the graph is isolated.
In the context of trees, being connected is crucial because:
  • This means if you start at any vertex, you can reach any other vertex following the edges of the graph.
  • A tree is a type of connected graph that also possesses no cycles.
  • The property of being connected without forming cycles lends trees their unique structure, allowing for a single path between any two vertices.
Acyclic Graph
An acyclic graph is one that does not contain any cycles. In simpler terms, you can't start from one vertex, traverse along the edges, and return back to the starting vertex without retracing steps.
For trees, the acyclic nature ensures:
  • When two different paths are assumed between two vertices, they would form a cycle, which can't happen in a tree.
  • This characteristic helps trees to maintain a single path between any two nodes, as cycles would otherwise create multiple paths.
  • The absence of cycles allows trees to be predictable and manageable in navigation and computation tasks.
Unique Path
The unique path property of trees is one of its most defining characteristics. This means that only one path exists between any pair of vertices.
Here's why this is significant:
  • This comes from the tree's properties: being both connected and acyclic.
  • If, for any two vertices, two different paths existed, it would form a cycle, which contradicts the tree's acyclic nature.
  • The uniqueness of paths makes trees invaluable for various computer science applications, like network routing and hierarchy representation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

When will the adjacency matrix of a graph be symmetric? Of a digraph?

A tree with exactly two vertices of degree 1 must be a path.

Show that in a network \(G\) with all \(c_{i j}=1\), the maximum flow equals the number of edge-disjoint paths \(a \rightarrow t\)

Sketch the graph whose adjacency matrix is: $$\left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right]$$

Show that a graph \(G\) with \(n\) vertices can have at most \(n(n-1) / 2\) edges, and \(G\) has exactly \(n(n-1) / 2\) edges if \(G\) is complete, that is, if every pair of vertices of \(G\) is joined by an edge. (Recall that loops and multiple edges are excluded.
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Famous Physicists

Read Explanation

Force

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Measurements

Read Explanation

Electromagnetism

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.