Hi Larry

A star orbitting a planet isn't going to happen in any plausible natural arrangement of matter. You could make a 'planet' of pure iron that masses up to 1.4 solar masses before it implodes past the Chandrasekhar limit into a neutron star. Lighter elements have smaller limiting masses before pycnonuclear reactions in their gizzards start fusing elements - a 0.3 solar mass helium 'planet' would become a helium star, for example. Carbon and oxygen would become unstable to fusion at around 0.9 solar masses.

Just how any natural process fractionates out enough pure element, from a stellar birth nebula, to do this is unknowable. Perhaps a super-planet with orbital dwarf stars would be an unmistakeable bit of sidereal engineering, perfect for CETI.

Alternatively it could be made as an ultra-light shell and deliberately kept larger than the Jeans radius so it doesn't gravitationally collapse. What material would be strong enough is currently unknown.

Much better would be a cloud of stably orbitting habitats made of diamond - real estate and it'd stand out against the sky.

Adam