graphs and networks - How to find all vertices reachable from a start vertex following directed edges?

How to find all vertices reachable from a given start vertex following directed edges, in a cyclic directed graph given as

Graph[{v1, v2, ...}, 
      {v1 -> v11, v1 -> v12, ..., v2 -> v21, v2 -> v21, ..., vn -> vn1, vn -> vn2, ...}]

where all the ending vertices of edges vij are in the list of vertices {v1, v2, ...}. ?

Answer

One can use VertexOutComponent[] to find all the vertices connected to a given vertex in a directed graph:

In[107]:= edges={1->3,1->4,2->4,2->5,3->5,6->7,7->8,8->9,9->10,6->10,1->6,2->7,3->8,4->9,5->10};
In[114]:= vertices=Sort@DeleteDuplicates[Flatten[List@@@edges]];
In[115]:= g=Graph[vertices,edges];
In[116]:= {#,VertexOutComponent[g,{#}]}&/@vertices//Grid
Out[116]= 1 {1,3,4,5,6,7,8,9,10}
2   {2,4,5,7,8,9,10}
3   {3,5,8,9,10}
4   {4,9,10}
5   {5,10}

6   {6,7,8,9,10}
7   {7,8,9,10}
8   {8,9,10}
9   {9,10}
10  {10}

It should work for any directed graph whether it's acyclic or not. The analogue of VertexOutComponent[] for undirected graphs is ConnectedComponents[].