Final answer:
To confirm that the complement of any spanning tree in a connected graph with a cycle includes at least one edge from the cycle, we recognize that adding all edges of a cycle would introduce a cycle in the spanning tree, which contradicts its acyclic nature.
Step-by-step explanation:
The subject of the question is mathematics, specifically within the field of graph theory. To show that the complement of any spanning tree of a connected graph G with at least one cycle includes at least one edge from a cycle C, consider the following:
- Firstly, a spanning tree T of a connected graph G includes all vertices of G, but has no cycles and therefore has exactly n-1 edges, where n is the number of vertices in G.
- Since C is a cycle in the graph G, it must contain at least one edge that is not in T, as inclusion of this edge in T would create a cycle, which contradicts the definition of a tree.
- If we were to remove edges of C from the graph G until it no longer forms a cycle, we can integrate those edges into T one by one without creating a cycle until the last edge. Adding this last edge from C would create a cycle, therefore, this edge cannot be part of T and must be in the complement of T.
Therefore, the complement of any spanning tree of G must always include at least one edge from any cycle C within the graph.