Is the graph acyclic?
Mathematica, 80 44 40 30 22 bytes

#//.x_:&gt;#[[Union@@x]]&amp;


Uses 1-based indexing for input. Returns an empty list iff the graph is acyclic.

Example:

In[1]:= #//.x_:&gt;#[[Union@@x]]&amp;[{{2,3},{4,5},{6},{},{7},{7},{}}]

Out[1]= {}

In[2]:= #//.x_:&gt;#[[Union@@x]]&amp;[{{2,4},{3},{1},{2}}]

Out[2]= {{2, 4}, {3}, {1}, {2}}

In[3]:= #//.x_:&gt;#[[Union@@x]]&amp;[{{1}}]

Out[3]= {{1}}