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Chapter 12: Problem 12

\([\mathrm{BB}]\) Suppose \(\mathcal{P}_{1}\) and \(\mathcal{P}_{2}\) are two paths from a vertex \(v\) to another vertex \(w\) in a graph. Prove that the closed walk obtained by following \(\mathcal{P}_{1}\) from \(v\) to \(w\) and then \(\mathcal{P}_{2}\) in reverse from \(w\) to \(v\) contains a cycle.

Short Answer

Expert verified
The closed walk contains a cycle due to overlapping vertices in \( \mathcal{P}_{1} \) and \( \mathcal{P}_{2} \).

Step by step solution

01

Understand the Concepts

The problem involves two paths, \( \mathcal{P}_{1} \) and \( \mathcal{P}_{2} \), from vertex \( v \) to vertex \( w \). A closed walk means starting and ending at the same vertex, here \( v \), after moving through both paths.
02

Define Paths and Reverse

Consider path \( \mathcal{P}_{1} = (v, x_1, x_2, \ldots, w) \) and path \( \mathcal{P}_{2} = (v, y_1, y_2, \ldots, w) \). The reverse of \( \mathcal{P}_{2} \) is \( (w, y_k, \, ..., \, y_1, v) \).
03

Form the Closed Walk

Concatenate \( \mathcal{P}_{1} \) followed by the reverse of \( \mathcal{P}_{2} \) to form the closed walk: \( (v, x_1, x_2, \ldots, w, y_k, \ldots, y_1, v) \).
04

Identify Overlapping Vertices

Since both paths start at \( v \) and end at \( w \), and no particular restrictions are set on vertices between \( v \) and \( w \), it is likely that these two paths share at least one vertex besides \( v \) and \( w \).
05

Detect the Cycle

The first repeated vertex in \( \mathcal{P}_{1} \) and the reverse of \( \mathcal{P}_{2} \) defines a subpath in the walk that forms a cycle. This occurs because the closed walk retraces itself after reaching \( w \) through different paths.

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Key Concepts

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

Paths in Graphs
In graph theory, a **path** is a sequence of edges connecting a sequence of vertices. Imagine a map in which you trace a route from one landmark to another without retracing your steps. In mathematical terms, a path in a graph is denoted as a sequence of vertices
  • For example, a path \( \mathcal{P}_1 = (v, x_1, x_2, \ldots, w) \) means starting from vertex \(v\), passing through vertices \(x_1, x_2, \ldots\), and ending at vertex \(w\).
  • Paths can be directed or undirected, and they can have varying lengths. \( \mathcal{P}_1 \) and \( \mathcal{P}_2 \) illustrate different potential routes between \(v\) and \(w\).
Paths help in exploring the connections available within a graph. They serve as building blocks for other important concepts like cycles and closed walks, and are fundamental in algorithms for finding shortest paths or connectivity.
Cycles in Graphs
A **cycle** in a graph is a path that starts and ends at the same vertex with no other repetitions of vertices along it. Consider it like a loop in a race track鈥攜ou begin and end at the starting line without repeating any section of the track.
  • In any cycle, every edge and every vertex (except the starting and ending vertex) is visited exactly once.
  • In our exercise, proving the existence of a cycle involves showing that after traversing through two different paths \( \mathcal{P}_1 \) and reversed \( \mathcal{P}_2 \), some vertices are revisited. These intersections indicate the presence of a cycle.
Finding cycles in graphs is important in detecting redundancies and potential issues in networks, such as data flow loops or circular dependencies in programming.
Closed Walks
A **closed walk** in a graph returns to the starting vertex, but unlike cycles, vertices and edges can be repeated. It's similar to taking a stroll around your neighborhood and ending back home, possibly taking the same street more than once.
  • In the problem at hand, when paths \( \mathcal{P}_1 \) and the reverse of \( \mathcal{P}_2 \) are combined, the sequence forms a closed walk because it starts and finishes at the vertex \(v\).
  • The closed walk includes higher flexibility than cycles because paths repeat vertex visits when necessary, offering insights into structural redundancies in complex graph networks.
Understanding closed walks is crucial in areas like computer network design and transportation systems, where circular routes can represent both conveniences and challenges.

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Most popular questions from this chapter

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