The fine k-curve graph is hyperbolic; the finitary curve graph has diameter 2, a contractible flag complex, all countable graphs as induced subgraphs, and automorphism group equal to the homeomorphism group of the surface.
On the complexity of finite subgraphs of the curve graph
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Hyperbolicity, topology, and combinatorics of fine curve graphs and variants
The fine k-curve graph is hyperbolic; the finitary curve graph has diameter 2, a contractible flag complex, all countable graphs as induced subgraphs, and automorphism group equal to the homeomorphism group of the surface.