For fixed t, both C_t and Θ_t have the induced Erdős–Pósa property for induced minors with hitting set size O(tk log k), implying O(tk log k)-dominated balanced separators and a QPTAS for MWIS in kΘ_t-induced-minor-free graphs.
Finding large sparse induced subgraphs in graphs of small (but not very small) tree-independence number
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Induced Erd\H{o}s--P\'osa property for long holes, long thetas, and beyond
For fixed t, both C_t and Θ_t have the induced Erdős–Pósa property for induced minors with hitting set size O(tk log k), implying O(tk log k)-dominated balanced separators and a QPTAS for MWIS in kΘ_t-induced-minor-free graphs.