H-free Subdivision is polynomial-time solvable when every component of H is a subdivided star or a subdivided bistar with at most one bistar, and NP-complete with no 2^{o(k)} n^{O(1)} algorithm under ETH for H satisfying any of seven listed structural conditions.
Contracting edges to destroy a pattern: A complexity study
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On the complexity of edge subdivision to $H$-free graphs
H-free Subdivision is polynomial-time solvable when every component of H is a subdivided star or a subdivided bistar with at most one bistar, and NP-complete with no 2^{o(k)} n^{O(1)} algorithm under ETH for H satisfying any of seven listed structural conditions.