16.8: Exercises
Prove that if a graph contains a closed trail then it also contains a proper cycle.
Spanning trees.
For each of the graphs in Exercises 2–3, draw a spanning tree by inspection.
Reducing to a spanning tree.
For each of the graphs in Exercises 4–5, use the following algorithm to obtain a spanning tree.
- If the graph contains a proper cycle, remove one edge of that cycle.
- If the resulting subgraph contains a proper cycle, remove one edge of that cycle.
- If the resulting subgraph contains a proper cycle, remove one edge of that cycle.
- etc..
- Continue until there are no proper cycles left.
Depth-first and breadth-first spanning trees.
For each of the graphs in Exercises 6–8, determine both a depth-first and breadth-first spanning tree. Use any vertex you like as the starting node.