First Graph

The given graph has the edges: [ [a, b], [a, e], [a, f], [b, c], [b, d], [b, e], [b, f], [d, e], [e, f] ]

To form a spanning tree, we need to select a subset of edges that includes all vertices and connects them without creating any cycles. The spanning tree must have $n - 1$ edges, where $n$ is the number of vertices. Since we have 6 vertices $\{a, b, c, d, e, f\}$, the spanning tree will have 5 edges.

One possible spanning tree is:

[ [[a, b], [b, c], [b, d], [a, e], [a, f]] ] This connects all vertices and avoids cycles.

Second Graph

The given graph has the edges: [ [a, d], [a, f], [b, d], [b, e], [b, f], [c, e], [c, f], [d, e] ]

Again, we need a spanning tree with 5 edges connecting all 6 vertices without cycles.

One possible spanning tree is:

[ [[a, d], [b, d], [b, e], [c, e], [a, f]] ] This also connects all vertices and avoids cycles.

---

Would you like further details on how the spanning trees were selected or on any related topic?

5 Related Questions:

1. What is the definition of a spanning tree?
2. How do you determine if a graph has a cycle?
3. How many spanning trees can a complete graph have?
4. What is Kruskal's algorithm for finding a minimum spanning tree?
5. What is the difference between a minimum spanning tree and a regular spanning tree?

Tip:

When constructing spanning trees, ensure you avoid cycles by carefully adding edges and checking connectivity.