Some gardeners who are short on in-ground growing space like to graft a potato and tomato plant together to grow one plant in a single pot which produces both tomatoes and potatoes. While this process is possible because both parent plants are of the same plant family, it isn't ideal for growing your biggest crop. If you have the space, standard growing of individual plants is recommended.
Move the pots of your tomato and potato plants next to one another. Bring the main stems of each plant together halfway up the height of the plants. Tie the stems together with a soft string or plant ties tight enough to hold them against one another without cutting into the stem.
Use a sharp knife to cut a vertical sliver from the tomato plant's stem just above the string or tie between the two plants. The sliver should be about an inch long and cut into the thickness of the stem of the plant only by a third. Cut an identical sliver out of the potato's main stem between the two plants so the two slivers are paired together.
Press the cut-out sliver areas together and tie them with string or plant ties to hold them against each other. This tie can be tighter than the one below it so the cut surfaces can begin to graft together. Wrap a strip of grafting tape around your string to seal off the cut area entirely.
Leave the plants alone over the next week other than watering both pots as needed. Signs of wilting or yellowing can signal that the graft isn't working. Look for both plants to appear healthy and possibly show new growth, as well as a bulge to appear around the grafted area. If you aren't seeing this within the first week, allow another week to pass as the plants blend together.
Complete the graft only after at least a week has passed and both plants appear healthy. Do so by cutting off the top of the potato plant an inch above the graft site with a single, clean slice and cut off the bottom of the tomato plant just below the graft as well. Let the fresh cuts heal over for a day or two before removing the grafting tape and strings.
Add a garden stake to the potato's original pot and support the upper half of the plant by tying it to the stake. Water and care for your plant as you would any other potted plant and harvest tomatoes when they are ripe. The potatoes should be ready for harvest three weeks after the upper portion of the plant dies off in the fall.