Question

Let \(D\) be a directed graph with an outdegree of each vertex of at least one. Prove that \(D\) contains a directed cycle. Show that the same result holds if the hypothesis is that each node has an indegree of at least one.

Short answer

A directed cycle exists in either scenario by tracing a path indefinitely, assuring repetition and thus a cycle.

Step-by-step solution

Understand the Problem

We are given a directed graph, where each vertex has an outdegree of at least one. This means that from each vertex, there is at least one outgoing edge. Similarly, we need to show that if each vertex has an indegree of at least one, the graph also contains a directed cycle.

Prove the Cycle with Minimum Outdegree 1

Start from an arbitrary vertex. Since each vertex has an outdegree of at least 1, continue to follow the outgoing edges from each reached vertex. Since the graph is finite, repeating vertices must eventually occur, creating a cycle.

Formalize the Argument for Step 2

By following edges, the path must either visit vertex previously visited (form a cycle) or continue indefinitely. Since there is a finite number of vertices, the latter is impossible, thus a cycle exists.

Prove the Cycle with Minimum Indegree 1

Choose any vertex and trace edges backwards, starting from choosing backward edges. Since each vertex has an indegree of at least 1, there is always an incoming edge to follow. Eventually, some vertices must be repeated, forming a cycle.

Formalize the Argument for Step 4

If we cannot keep finding new vertices going backwards, a cycle must be reached because of the finite number of vertices, as the repeated visit to a vertex means a cycle.

Key concepts

Directed Graph

In graph theory, a [directed graph](https://en.wikipedia.org/wiki/Directed_graph), or digraph, is a set of vertices connected by edges, each edge having a direction. This means that each connection between two nodes has a defined direction from one node to another. Unlike undirected graphs, where edges simply connect two nodes, directed graph edges have a clear start and end. These are represented by arrows in graphical representations, indicating the direction of the connection.

Directed graphs can model a variety of real-world situations, such as traffic networks, social media interactions, or task scheduling. Each vertex in a directed graph can have edges pointing to other vertices (outgoing) or edges coming from other vertices (incoming). This distinction between the direction of edges is crucial for understanding concepts like indegree and outdegree, which we will explore next.

Understanding directed graphs is fundamental to tackling problems involving paths, connectivity, and cycles, central themes in graph theory.

Indegree and Outdegree

The concepts of indegree and outdegree are critical when working with directed graphs. **Indegree** refers to the number of edges coming into a vertex. Meanwhile, **outdegree** is the number of edges leaving a vertex. These two measures offer insight into the structure and behavior of the graph.

For example:

• A vertex with an indegree of zero is a starting point with no incoming connections.
• A vertex with an outdegree of zero is an endpoint, with no edges leading away from it.

Understanding these two concepts is essential for deducing many properties of a graph, such as its likelihood to contain a cycle. If every vertex in a graph has at least one outgoing edge (outdegree ≥ 1), or at least one incoming edge (indegree ≥ 1), then this condition plays an important role in cycle detection. The basic principle is simple: paths must eventually revisit vertices in a finite graph, thus forming cycles.

Directed Cycle

A directed cycle is a path in a directed graph that starts and ends at the same vertex with all edges pointing in a consistent direction along the path. Directed cycles are a significant aspect of graph theory because they indicate repeated processes or feedback loops within systems.

Why do directed cycles matter?

• They can help identify processes that might repeat indefinitely, such as loops in computer programs.
• In business workflows, cycles can signify inefficiencies or redundancy.

In our exercise, proving the existence of a directed cycle involves following edges consistently according to the conditions set: either every vertex has an outdegree of at least one or an indegree of at least one. Given the finite number of vertices, we must eventually revisit some vertices creating a cycle. Understanding these patterns helps in optimizing and predicting the behavior of networks modeled as directed graphs.

Graph Traversal

Graph traversal is the process of visiting vertices in a graph in a systematic manner. In directed graphs, traversal follows the direction of the edges. This is key in exploring and analyzing the pathways and connectivity between nodes in the graph.

There are several strategies for graph traversal, such as:

• Breadth-First Search (BFS): Traverses the graph level by level, using a queue to explore each vertex and its neighbors before moving deeper.
• Depth-First Search (DFS): Uses a stack approach to explore as far as possible along each branch before backtracking.

These methods are useful for finding connected components, shortest paths, or verifying the presence of cycles like those discussed in our exercise. When applying these methods, especially in directed graphs with constraints on indegree and outdegree, the paths formed during traversal reveal much about the graph's structure — confirming how cycles can be detected and why every vertex inevitably initializes or participates in these cycles.