Relative center construction for $G$-graded C${}^{\ast }$-tensor categories and Longo-Rehren inclusions

Gelaki-Naidu-Nikshych and Turaev-Virelizier showed the existence of $G$-braiding on the relative Drinfeld center of a $G$-graded tensor category. We will explain this concept from the viewpoint of Longo-Rehren inclusions.