If a graph is labelled as a forest it does not contain any cycles, meaning it consists of all trees, which I realize can even be a single node (since that is technically a tree).
If a graph is labelled as a spanning forest, it is a forest that contains every vertex of G such that two vertices are in the same tree of the forest when there is a path in G between these two vertices.
Aren't these basically the exact same? I am having a bit of trouble telling the difference between the two.
Best Answer
So suppose I have three disjoint sets of vertices: $\{v_{1}\} \cup \{v_{2}\} \cup V(C_{3})$. Here, $\{v_{1} \} \cup \{v_{2}\}$ is a forest which does not span, while $\{v_{1}\} \cup \{v_{2}\} \cup (C_{3} - e)$ is a spanning forest, for $e \in E(C_{3})$.