@gabrieljim The vertex in your example even seems to be an 8-pole. In any case, you could select that pole-vertex and use “Rip and Fill” (Alt + V):

1) Place the mousepointer on the side to which you want to extrude that vertex:

2) Now press Alt + V and drag the newly created vertex to its final position:

3) You can now select the edge between the two adjacent center triangles in “Edge Select Mode” and dissolve it with x + i:

