Complexity of 3-connected 3-augmentations under fixed embedding and/or 2-connected input

Classify the computational complexity of deciding whether a subcubic planar graph admits a 3-connected 3-augmentation in the remaining open settings: (i) when the planar embedding of the input graph is fixed, and/or (ii) when the input graph is restricted to be 2-connected.

Background

The paper proves NP-completeness for deciding the existence of a 3-connected 3-augmentation in the variable-embedding setting, even for connected inputs. All cases with connectivity target k ≤ 2 are solvable in polynomial time, for both fixed and variable embeddings.

The authors highlight that the cases involving 3-connectivity remain unresolved when either a fixed embedding is required or the input graph is constrained to be 2-connected (or both). They conjecture these cases are also NP-complete, but a proof is not yet known.