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When all preparation is ready, i select the grafting stock plants and start the fun. First i do horizontal cut about 3-4 inch high on the stock or very tip of it, just below the growing point, for the seedlings. I next bevel the edges of my stock under about 45 degree angle, then clean my knife with a rag soaked in rubbing alcohol, then after i make another thin horizontal cut, only 1/8 of an inch or so and i leave this slice on top to prevent the lower surface from drying out. After i clean my grafting knife again i cut off my scion from mother plant, bevel the edges, similar to the way i did it on the grafting stock, one more straight cut and after removing the protective slice on the stock i join the two plants together in the way so the vascular rings match each other or overlap. Then i place my rubber bands over the stock and over the scion. 

There are only few ways of propagating Cacti.

The most common way is by seeds, some species by cuttings and offsets, but there is also grafting. Grafting can be done to rescue a damaged or dying plant, to solve problems of structure growth, conditions or disease and of course as a propagating technique to increase number of plants. Also there is a technique  of grafting to gain time, when very slow growing species are grafted for some time and then takes off the stock and continue to grow on their own roots as a mature specimen. Grafting itself can be done in a few different ways, but the most [Надпись: Grafted Mammillaria and Lobivia varieties on Cereus peruvianus]  popular and efficient is flat-grafting, that used for seedlings and cuttings alike. For the last 30 years or so i have been using this technique with grafting all kinds of species of cacti and i have arrived at what i think is the most successful method of propagation of cactus plants. I do prefer flat-grafting using seedlings of any given cactus species from a couple of months to a year old that will be grafted on the stock with a small diameter of vascular ring. I found that Lemairocereus pruinosus, Myrtillocactus geometrizans, Echinopsis subdenudata, Eriocereus pomanensis, Peireskiopsis sphatulata and Hylocereus varietis are to be very good for this purpose. A couple words about tools: i use a very simple set, one grafting knife, the type that made by Olfa, heavy duty with snap-off retractable blades, rubbing alcohol to clean the blade, a piece of old towel to wipe the knife and rubber bands, different straight to hold the scion on the stock. Before grafting to take  place i usually prepare the stock plants and the scion plants, offsets, pops or seedlings it make s no difference, they  need to be ready. Because the stock must be actively growing, it is better to give the stock a good drink of water a couple of days before. Depending on what type of scion you have in mind the preparation of scion will be similar or completely different. If you are working with grafting offers or pops you probably need to water that plant 2-3 days before grafting will take place, just like you did with the grafting stock. if you have to work with seedlings (also actively growing) i would recommend you to remove seedlings from the soil, wash the roots under water and then let them dry out for a few hours.
Normally it takes 2-3 rubber bands to secure the scion on top of the stock. If you are working with a seedling then one or maybe two rubber bands are needed. I use different size rubber bands for different size of the scion to apply optimum pressure. The rubber bands exert constant pressure, which can be increased or decreased, depending on how far rubber bands are stretched. Attaching the scion on top of the grafting stock is the most difficult part of the grafting and must be done as fast as possible.                              
Now for the aftercare of the graft. because the plant needs humidity and some warmth, i put all my newly grafted plants in a shaded spot in the greenhouse, out of direct sunlight, sometimes even under the bench. Remember not to water the plants at this stage. Depending on the size of the scion and the pressure of the rubber bands, [Надпись: Strombocactus disciformis, grafted on Lemairocereus pruinosus stock, about 3-4 years old and over 2 inch in diameter.]  I keep them on somewhere from two to seven days. As soon as the exposed cut surface of the stock plant has dried it is safe to water the plants and expose them to normal light. Check the plants about once a week or so and remove any offsets from the grafting stock, as it needs to concentrate all its vigor for the scion. After a few weeks it is all right to fertilize the stock and the scion, by this time you should seen new growth on the scion. 
After the first season it can reach an amazing size. Some of the varieties can double or triple in the very first year. Of course some of my favorite drafts are different varieties of Ariocarpus, that naturally are very slow growing plants and even they can't  resist and grow very nicely after being grafted as a seedlings. By the end of second year some of my plants have reached a very nice size, as much as 3-4 inch in diameter and most of them will flower in the second year, after their first dormancy! The number of flowers also will surprise you a lot, because they are far more numerous then when the plants are on their own roots!
There are some collectors who do not like grafted plants because they look unnatural. I believe most of them have never attempted a single graft, if they would try it, perhaps they will chance their mind. Grafting is not an end in itself, but more of an intermedium stage of plant growing and a way to increase what one already has. It is of course, possible to make your grafted plants look like they have never see the grafting stock. For that purpose (after you have well-established, flowering size plant) you can cut the stock an inch below the scion and use rooting hormone to re-root it. After a year or even sooner  no one will tell if this plant was grafted or not.