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. A closed walk in a graph means that you start and end at the same vertex.
- 2. All cycles are closed walks, but not all closed walks are cycles since they can revisit vertices.
- 3. The presence of a closed walk in an undirected graph implies that there is at least one cycle within the graph.
- 4. Eulerian circuits are examples of closed walks that visit every edge exactly once.
- 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?
"Closed" appears in:
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
