$h$-vectors of edge rings of odd-cycle compositions

Let $\mathbb{K}[G]$ be the edge ring of a finite simple graph $G$. Investigating properties of the $h$-vector of $\mathbb{K}[G]$ is of great interest in combinatorial commutative algebra. However, there are few families of graphs for which the $h$-vector has been explicitly determined. In this paper, we compute the $h$-vectors of a certain family of graphs that satisfy the odd-cycle condition, generalizing a result of the second and third named authors. As a corollary, we obtain a characterization of the graphs in this family whose edge rings are almost Gorenstein.