Here is the full list of all 12 Vertex Dispenser achievements.
Complete Episode 1.
Complete Episode 2.
Complete Colour Puzzle.
Collect the Expand Edges ability on Time to Defend (normal difficulty).
Eliminate the Enemy on Mythic Portal (normal difficulty).
Complete Mythic Portal in 1:30 (normal difficulty).
Complete Slaughterhouse without dying (normal difficulty).
Complete Arcane Lore (maximum difficulty).
Exceed seven colours.
Have two Doppelgängers in play at once.
Have a dozen white vertices.
Beat m hard AIs with n human players, where m>n.