What is the difference between a forest and a spanning forest? 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.

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 $e \in E(C_{3})$.