We characterize the well-covered property for theta-ring and ring graphs. Furthermore, we prove that Cohen–Macaulayness, pure shellability and pure vertex decomposability are equivalent for theta-ring graphs. Also, we give a combinatorial characterization of these graphs.
"Well-covered and Cohen–Macaulay theta-ring graphs." J. Commut. Algebra 13 (4) 461 - 477, Winter 2021. https://doi.org/10.1216/jca.2021.13.461