The kind of star that becomes a giant when fusion slows down, the outer atmosphere expands, luminosity increases, and it begins to fuse helium into carbon is a main sequence star.

Here's why:

* Main Sequence Stars: These are the most common type of stars, like our Sun. They spend the majority of their lives fusing hydrogen into helium in their core.

* Fusion Slowdown: As hydrogen fuel in the core dwindles, the rate of fusion slows down. This causes the core to contract under its own gravity.

* Expansion and Luminosity: The contraction of the core heats up the outer layers of the star, causing them to expand significantly. This expansion increases the star's surface area, leading to a dramatic increase in luminosity.

* Helium Fusion: The increased temperature and pressure in the core eventually become sufficient to ignite helium fusion. This process creates carbon as a byproduct.

Therefore, it's the main sequence stars that undergo this transformation into giant stars.