Graphs Without Star Cutsets

Is there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no star cutset?

Contributed by Bruce Reed

Conjecture If neither $G$ nor $G^c$ has a star cutset then the disk-structure of $G$ is connected

(A disk is a hole or an antihole. Two disks are adjacent in the disk structure if they share at least $2$ vertices).

Contributed by Ryan Hayward

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