Empty Graph
Root nodes: 

Graph with 0 edges
Root nodes: 0 1 2 3 4

Graph with 1 vertex
Root nodes: 0

Undirected Graph
Root nodes: 0 3

Directed Graph out trees
Root nodes: 0 3

Directed Graph in trees
Not a forest.

Undirected Graph with cycle
Not a forest.

Undirected Graph with ecount>vcount-1
Not a forest.

Undirected Graph with self-loop
Not a forest.

Directed Multigraph
Not a forest.

Disjoint union of a cycle with two trees, directed
Not a forest.

Disjoint union of a cycle with two trees, undirected
Not a forest.

