Find all reachable nodes in a graph
Mathematica 63 40 37
My earlier submission was incorrect. I had misinterpreted what ConnectedComponents returns.  The present approach relies on Belisarius' suggestion to useVertexOutComponent. 3 additional chars saved by alephalpha.
The following generates an 11 by 11 adjacency matrix of 0's and 1's, where 1 indicates that the vertex with the row number is connected directly to the vertex corresponding to column number of the cell. (SeedRandom ensures that your matrix a will match mine.)
SeedRandom[21]
a = Array[RandomChoice[{0.9, 0.1} -&gt; {0, 1}] &amp;, {11, 11}];


Example
(37 chars)
 AdjacencyGraph@a~VertexOutComponent~4


{4, 1, 7, 8, 9, 3}


Verifying
The directed graph displays how to reach vertices {1, 7, 8, 9, 3} from vertex 4.
AdjacencyGraph[a, ImagePadding -&gt; 30, VertexLabels -&gt; &quot;Name&quot;]