Proofs

Graph:

             [0]
              |
              v
             [1]
              |
              v
             [2]-. 
              |   \
              v    \
             [3]   [6]
              |     |
              v     v
             [4]   [7]
              |     /
              v    /
             [5]<-´

dom(s) = { s }

Obvious

All vertices 'w belonging to V' are dominated by s

Follows from dominance definition. If all paths from s to w contain v then v _>>_ w

Dominance is reflexive

This follows directly from the definitions since any path (w0, w1, w2, ..., wk) from s = w0 to w=wk includes w.