The $h$-vectors of toric ideals of odd cycle compositions revisited

Let $G$ be a graph consisting of $s$ odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the $h$-polynomial for the quotient ring $R/I_G$, where $I_G$ is the toric ideal of $G$, in terms of the number and sizes of odd cycles in the graph. The purpose of this note is to prove the stronger result that these toric ideals are geometrically vertex decomposable, which allows us to deduce the result of Bhaskara, Higashitani, and Shibu Deepthi about the $h$-polyhomial as a corollary.