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Chapter 10: Problem 59

Let \(P_{1}\) and \(P_{2}\) be two simple paths between the vertices \(u\) and \(v\) in the simple graph \(G\) that do not contain the same set of edges. Show that there is a simple circuit in \(G\) .

Short Answer

Expert verified
Two simple paths with different edges imply a different traversal possibility, forming a simple circuit in G.

Step by step solution

01

Understand the Definitions

A simple path is a path in a graph that does not repeat vertices. A simple circuit (or cycle) is a closed path with no repeated vertices except for the starting and ending vertex.
02

Examine Paths P1 and P2

Consider the two simple paths, P_{1} and P_{2}, that start at vertex u and end at vertex v . Since P_{1} and P_{2} do not contain the same set of edges, they must traverse different routes between u and v .
03

Identify Shared and Non-Shared Edges

Denote the sets of edges in P_{1} and P_{2} as E_{1} and E_{2} respectively. Since E_{1} eq E_{2}, there exist at least one edge in E_{1} that is not in E_{2} and vice versa.
04

Construct a Simple Circuit

Start from the first vertex u . Follow P_{1} until you reach the first edge that is not in P_{2}. From this point, switch to P_{2} and continue to v . Finally, follow P_{2} in reverse until you return to the starting vertex u .
05

Verify the Simple Circuit

This traversal from u following P_{1}, switching to P_{2}, and returning results in a closed path with no repeated vertices (except u ), hence forming a simple circuit.

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Key Concepts

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

Simple Path
A simple path is a fundamental concept in graph theory. It is a path that connects a sequence of vertices without revisiting any vertex. Think of it like taking a walk in a park where you step on each tile only once until you reach your destination. This ensures no loops exist within the path.
For example, if you have vertices labeled A, B, C, and D, a simple path could be A -> B -> C. Notice that once you step on B and move to C, you don't step on B or A again.
Simple paths are crucial for many algorithms in graph theory, primarily because they prevent repetitive traversal, keeping computations efficient. Understanding simple paths helps in constructing more complex structures in graphs.
Simple Circuit
A simple circuit, or cycle, is a path that starts and ends at the same vertex without repeating any other vertex along the path. Imagine looping through the park mentioned earlier, returning to your starting point without retracing any steps, except the last step that brings you back to start.
For instance, if vertices A, B, C, and D form a cycle, a simple circuit could be A -> B -> C -> D -> A. Each vertex is visited exactly once before returning to the starting point A.
Simple circuits are important for detecting cycles in graphs, which has applications in network theory, biology, and computer science. Cycles can represent feedback systems, loops in programming, and repetitive processes.
Graph Traversal
Graph traversal refers to the process of visiting, checking, and updating each vertex in a graph. Two primary methods of traversal are Depth-First Search (DFS) and Breadth-First Search (BFS).
In DFS, you start at a vertex and explore as far as possible along each branch before backtracking. Think of it like exploring all corridors in a labyrinth before trying another route.
In BFS, you start at a vertex and explore its neighbors before moving to the next level of neighbors. It鈥檚 like a wave spreading outwards.
Efficient graph traversal is essential for solving many graph-related problems such as finding shortest paths, checking connectivity, and detecting cycles. Understanding how to traverse graphs helps in analyzing complex networks and systems.

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Most popular questions from this chapter

The intersection graph of a collection of sets \(A_{1}\) , \(A_{2}, \ldots, A_{n}\) is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph of these collections of sets. a) \(A_{1}=\\{0,2,4,6,8\\}, A_{2}=\\{0,1,2,3,4\\}\) \(A_{3}=\\{1,3,5,7,9\\}, A_{4}=\\{5,6,7,8,9\\}\) \(A_{5}=\\{0,1,8,9\\}\) b) \(A_{1}=\\{\ldots,-4,-3,-2,-1,0\\}\) \(A_{2}=\\{\ldots,-2,-1,0,1,2, \ldots\\}\) \(A_{3}=\\{\ldots,-6,-4,-2,0,2,4,6, \ldots\\}\) \(A_{4}=\\{\ldots,-5,-3,-1,1,3,5, \ldots\\}\) \(A_{5}=\\{\ldots,-6,-3,0,3,6, \ldots\\}\) c) \(A_{1}=\\{x | x < 0\\}\) \(A_{2}=\\{x |-1 < x < 0\\}\) \(A_{3}=\\{x | 0 < x < 1\\}\) \(A_{4}=\\{x |-1 < x < 1\\}\) \(A_{5}=\\{x | x > -1\\}\) \(A_{6}=\mathbf{R}\)

In a round-robin tournament the Tigers beat the Blue Jays, the Tigers beat the Cardinals, the Tigers beat the Orioles, the Blue Jays beat the Cardinals, the Blue Jays beat the Orioles, and the Cardinals beat the Orioles. Model this outcome with a directed graph.

Suppose that a connected planar graph has eight vertices, each of degree three. Into how many regions is the plane divided by a planar representation of this graph?

Fleury's algorithm, published in \(1883,\) constructs Euler circuits by first choosing an arbitrary vertex of a connected multigraph, and then forming a circuit by choosing edges are successively. Once an edge is chosen, it is removed. Edges are chosen successively so that each edge begins where the last edge ends, and so that this edge is not a cut edge unless there is no alternative. Prove that Fleury's algorithm always produces an Euler circuit.

For which values of n are these graphs regular? $$ \begin{array}{lllll}{\text { a) } K_{n}} & {\text { b) } C_{n}} & {\text { c) } W_{n}} & {\text { d) } Q_{n}}\end{array} $$
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