You can never omit a determiner (a, the, my, this, every, etc.) in front of a singular, countable noun. Such a word is necessary in every case.

The definite article must be used here "The potato is ....." The potato is a particular type of vegetable. "The cheetah is the fastest land animal" The cheetah is a particular type of animal.

Great question. If we are talking about the potato as one of all types of vegetable, we do need the definite article. In this way we are looking at the potato as an abstract category, a type, a species. You can also use the plural, as in "potatoes are the best vegetables." Most natural would simply be "potatoes are the best" or "potatoes are the best vegetable" (the second may look awkward grammatically, but is totally normal in real life). You can use words for vegetables without articles when you are speaking about them as food, however. When you do this, you are changing the noun from countable (twenty potatoes) to uncountable (a lot of potato). "Potato is delicious, as is carrot, fish, chicken, etc." Nevertheless, although we CAN do this, with vegetables as food, we rarely do. We usually use the plural countable version: "These potatoes are delicious, and so are the carrots" (even though there may be many potatoes in this particular bowl of mashed potatoes, for example). With meats, we always use the uncountable version when we think of it as food (e.g. fish, chicken, octopus, squid, etc.). I hope that helps!