A long time ago in a galaxy called the Milky Way.....
An alien race had its planet thrown out of its solar system. It is now a rogue planet. They survived for thousands of generations in underground artificial environments. Then they achieve FTL and begin colonizing solar systems. A thousand years later the population of their planet is mainly too stubborn to leave since their race began there.
Since the government resides on their home planet and it does not intend on leaving (I know the following scenario is unlikely, but bear with me) they have decided to commission a hydrogen fusion project. This will build a miniature star orbiting their world via artificial means. I am not sure how big this star could be (or rather, how small?) and want it to supply the entire world with sunlight in a 26 hour cycle.
How can I go about doing this and how large would the 'star' have to be? Since we are dealing with a highly advanced civ, they can compress matter until it begins to fuse. The biggest problem I can foresee is supplying it with fuel.
You can use any technology level you want and are free to use the handwavium that such an impractical situation deserves. However, I do want answers to be descriptive and go into detail.