2.2. Spanning Trees

Try these first

1. 1.

   What is a spanning tree?

   Hint

   Think about what it means to connect all points without returning to the same point.

2. 2.

   How many edges does a spanning tree with n vertices have?

   Hint

   Relate it to the number of vertices.

3. 3.

   What is the definition of a spanning tree?

   • A cycle in a graph
   • A connected
   • acyclic graph covering all vertices
   • A disconnected graph

   Hint

   Consider the properties of trees in graph theory.

4. 4.

   True or False: A minimum spanning tree for a graph can have cycles.

   • True
   • False

   Hint

   Think about the definition of a tree.

5. 5.

   Given a complete graph with 5 nodes and the following edge weights: (A-B: 2, A-C: 3, A-D: 1, A-E: 4, B-C: 5, B-D: 3, B-E: 2, C-D: 4, C-E: 1, D-E: 6), find the minimum spanning tree and compute its total weight.

   Hint

   Look for the smallest edges that keep the tree connected.

6. 6.

   Design a road restoration plan for a small town affected by a natural disaster. You have the costs of repairing each road between towns. Describe how you would apply Prim's or Kruskal's algorithm.

   Hint

   Outline the costs and visualize the connections before applying the algorithm.