Question

Prove that if \(C_{1}, C_{2}, \ldots, C_{k}\) is a sequence of cycles in a directed graph \(D\) such that every two consecutive cycles have at least one common vertex, then the subgraph determined by the union of these cycles is strongly connected.

Short answer

The subgraph formed by the union of cycles is strongly connected due to shared vertices allowing bidirectional paths.

Step-by-step solution

Define a Strongly Connected Graph

A directed graph is strongly connected if there is a path from any vertex to every other vertex in the graph. We need to show this property for the subgraph that is the union of the given cycles.

Examine the Shared Vertices

Given that every two consecutive cycles, say \(C_{i}\) and \(C_{i+1}\), share at least one vertex, let's denote this common vertex as \(v\). This means there exists a path from any vertex in \(C_{i}\) to \(v\) and from \(v\) to any vertex in \(C_{i+1}\). This connection holds for every pair of consecutive cycles.

Establish Paths Between All Cycles

Due to the shared vertices between each pair of cycles, one can travel from any vertex in a cycle \(C_{i}\) to any vertex in \(C_{i+1}\) through these shared vertices. Extend this argument to show a path from any vertex in \(C_{1}\) to any vertex in \(C_{k}\), passing through all intermediate cycles.

Prove Bidirectional Paths

For the subgraph to be strongly connected, we need paths in both directions. Due to the cyclic nature and shared vertices, we can construct a path back from a vertex in \(C_{k}\) to any vertex in \(C_{1}\) by reversing the previous argument, hopping backward through shared vertices.

Conclude Strong Connectivity

Because we have shown a path exists in both directions between any pair of cycles, and by extension between any pair of vertices in the subgraph, the subgraph is strongly connected.

Key concepts

Cycles in Directed Graphs

Cycles in directed graphs play a crucial role in understanding graph properties. A cycle in a directed graph is a path where the starting vertex and the ending vertex are the same, with all edges following the direction. These cycles help illustrate how vertices are interconnected.
In the given exercise, we have a sequence of cycles, each sharing at least one common vertex with the next. This overlapping makes it easier to establish connections across the entire graph.
Recognizing these shared vertices is key for understanding how cycles can contribute to stronger connectivity. Each cycle in the sequence furthers the reachability of vertices, influencing the overall structure of the graph.

Graph Connectivity

Graph connectivity refers to the idea of how easily vertices within a graph are linked. In a connected graph, there is a path between any two vertices. For directed graphs, we often refer to 'strong connectivity'. Strong connectivity means there are paths in both directions between any pair of vertices.
In our exercise, the entire sequence of cycles is analyzed to see if their union is strongly connected. By ensuring that each consecutive cycle shares a vertex with the next, a path can form not just within individual cycles but extends throughout the graph.
This bidirectional path formation is essential for the graph to be considered strongly connected.

Path in Graph Theory

A path in graph theory is a sequence of edges that connects vertices without revisiting any vertex. In a directed graph, each edge has a direction, dictating the path's permissible flow.
The exercise describes a situation where paths exist between cycles through shared vertices. These paths are used constructively to demonstrate that you can start at any vertex in one cycle and reach any vertex in subsequent cycles.
Paths allow for efficient navigation from one part of the graph to another, and by sharing vertices between cycles, it ensures smooth transition and connectivity across all parts of the graph, enforcing the concept of strong connectivity.

Directed Graphs

Directed graphs, also called digraphs, consist of vertices connected by directed edges. These edges have a specific direction, represented by an arrow, signifying a one-way relationship between vertices.
Unlike undirected graphs, directed graphs offer a more complex and rich way to model relationships, such as one-way streets or hierarchical structures.
In the exercise, the directed nature of the graph plays into how cycles can be used to enforce strong connectivity. It's the directionality that dictates how paths form and how connectivity is maintained. Understanding directed graphs is essential to solving problems related to strong connectivity and allows us to tackle more complex graph challenges efficiently.