Closed

Definition

A graph is "closed" if it contains a cycle that returns to the starting vertex. In other words, you can traverse the graph and end up where you started without retracing edges.

5 Must Know Facts For Your Next Test

1. 1. A closed walk in a graph means that you start and end at the same vertex.
2. 2. All cycles are closed walks, but not all closed walks are cycles since they can revisit vertices.
3. 3. The presence of a closed walk in an undirected graph implies that there is at least one cycle within the graph.
4. 4. Eulerian circuits are examples of closed walks that visit every edge exactly once.
5. 5. Identifying whether a graph has a closed walk can be crucial for solving problems related to network connectivity and redundancy.

Review Questions

• 1. What defines a walk as "closed" in a graph?
• 2. Can a path be considered a closed walk? Why or why not?
• 3. How does identifying closed walks help in understanding the structure of networks?

Related terms

Cycle:

A sequence of vertices starting and ending at the same vertex with no repeated edges or vertices except for the starting/ending vertex

Eulerian Circuit: A circuit that visits every edge of a graph exactly once and returns to the starting vertex

Hamiltonian Cycle: A cycle that visits every vertex in a graph exactly once and returns to the starting vertex