Prepare the grafting stock, which will be the root part of the graft, by cleanly slicing away the top of an upright pitaya. The amount of stock you leave between the ground and the graft depends on plant size and health, and your end goals for grafting.
Prepare the scion, which is the pitaya you wish to propagate, by cleanly cutting the bottom part of it so that it is flat and level.
Examine the cut sections on both the stock and the scion, and locate the growth rings of each. When grafting, these rings must contact one another so they can fuse together. If the rings are of different diameters, match up whatever portion of them is possible.
Place the scion on top of the stock with at least some portion of the rings in contact with one another, and secure with tape. Make sure there is solid contact between the two parts of the graft.
Continue normal care for the cactus as is for one to two months before removing the restraints.