Construct an endpoint gadget for planar B2-CPG recognition

Construct a planar graph F_2 such that, in every B_2-CPG representation of F_2, some vertex v has a representing path P_v with no free endpoint and containing an exposed interior point, thereby enabling the Section 4 reduction to establish NP-hardness of recognizing planar B_2-CPG graphs.

Background

The paper resolves the recognition complexity of planar B_k-CPG graphs for k=0 and k=1, while the corresponding case k=2 remains unresolved. The authors explain that their one-bend reduction from Planar Exactly 3-Bounded 3-Sat relies on an end-eating gadget F with a path having no free endpoint and an exposed interior point.

Extending the reduction to two bends would require a planar graph F_2 with the analogous property in every B_2-CPG representation. If such a gadget existed, replacing F by F_2 would yield the same reduction and settle the planar B_2-CPG recognition problem. The construction of this gadget is identified as the remaining obstacle.

References

The case $k=2$ remains open. The reduction of Section 4 would extend directly to two bends if a suitable endpoint gadget were available. Specifically, suppose that there exists a planar graph $F_2$ such that, in every $B_2$-CPG representation of $F_2$, some vertex $v$ has a path $P_v$ with no free endpoint and contains an exposed interior point. Then replacing the end-eating gadget $F$ in Section 4 by $F_2$ would reduce Planar Exactly 3-Bounded 3-Sat to recognizing planar $B_2$-CPG graphs, exactly as in the one-bend case. The remaining obstacle is therefore the construction of such a graph $F_2$.

— Resolving Two Open Problems of Planar $B_k$-CPG Recognition: $k=0,1$  (2610.05838 - Sun et al., 5 Oct 2026) in Section Conclusion