Hydrogen fusion continues in a shell surrounding the core. Eventually the core is compressed enough to start helium fusion.
A star falls of the main sequence when it has exhausted it's supply of hydrogen.
Because they have exhausted their supply of hydrogen in the core. They might reenter the main sequence later, but that will be using hydrogen in the shell (the branch phase) rather than the core.
It has exhausted it's supply of hydrogen, and depending on the mass of the star, has, started "burning" other element's.
The thing the "main sequence" stars have in common is that they get their energy from the fusion of hydrogen (hydrogen-1 is converted into helium-4).
A main sequence star gets its energy by fusing hydrogen-1 into helium-4.
The "main sequence" is the region (on the HR diagram) for stars which burn hydrogen-1. Once stars use up most of their hydrogen-1 (and have significant amounts of helium-4), they leave the main sequence.
Stars are said to be off the main sequence when they stop fusing hydrogen into helium.
Stars on the main sequence (like the sun) are fusing hydrogen into helium in their cores.
The defining characteristic of a main sequence star burns hydrogen to helium in its core.
The main sequence stars are stars that fuse hydrogen, so the stars that have left the main sequence are the ones that have basically run out of hydrogen. They are the Red Giant stars, Supergiant stars and White Dwarf stars.
All main sequence stars fuse hydrogen.
If they are converting hydrogen to helium, then they are on the main sequence. This can be confirmed with a spectral analysis