You can split a shared vertex into multiple vertices (one for each polygonal face that shares it) using the Detach command. You can then perform independent operations on the split edges.
Splitting a vertex that is shared lets you separate the edges of the faces so you can create an opening in a polygon mesh or divide the polygon mesh altogether.
To split a vertex
The Detach feature ignores selected vertices that are not shared.
The selected vertices get split into multiple separate vertices that are not shared.