You can Rip a vertex.
Select the vertex you want. Press V and pull. Two disconneded vertices will be created out of the original.
From the Blender Manual:
Rip creates a “hole” into a mesh by making a copy of selected vertices and edges, still linked to the neighbor non-selected vertices, so that the new edges are borders of the faces on one side, and the old ones, borders of the faces of the other side of the rip.
Rip will only work when edges and/or vertices are selected. Using the tool when a face is selected (explicitly or implicitly), will return an error message “Cannot perform ripping with faces selected this way” If your selection includes some edges or vertices that are not “between” two faces manifold, it will also fail with message “No proper selection or faces include”.
Alternatively you can use Rip Fill or Alt+V.
Rip fill works the same as the Rip tool above, but instead of leaving a hole, it fills in the gap with geometry.