The Parameterized Complexity of Geometric 1-Planarity

A graph is geometric 1-planar if it admits a straight-line drawing where each edge is crossed at most once. We provide the first systematic study of the parameterized complexity of recognizing geometric 1-planar graphs. By substantially extending a technique of Bannister, Cabello, and Eppstein, combined with Thomassen’s characterization of 1-planar embeddings that can be straightened, we show that the problem is fixed-parameter tractable when parameterized by treedepth. Furthermore, we obtain a kernel for Geometric 1-Planarity parameterized by the feedback edge number $\ell$. As a by-product, we improve the best known kernel size of $O((3\ell)!)$ for 1-Planarity and $k$-Planarity under the same parameterization to $O(\ell \cdot 8^{\ell})$. Our approach naturally extends to Geometric $k$-Planarity, yielding a kernelization under the same parameterization, albeit with a larger kernel. Complementing these results, we provide matching lower bounds: Geometric 1-Planarity remains \NP-complete even for graphs of bounded pathwidth, bounded feedback vertex number, and bounded bandwidth.

💡 Research Summary

The paper initiates a systematic parameterized‑complexity study of the geometric 1‑planarity recognition problem, i.e., deciding whether a given graph admits a straight‑line drawing in which each edge is crossed at most once. The authors extend the structural‑parameter techniques of Bannister, Cabello, and Eppstein (originally developed for topological 1‑planarity) by exploiting Thomassen’s characterization of straight‑enableable 1‑planar embeddings (absence of B‑ and W‑configurations).

The first major contribution is an FPT algorithm when the input graph has treedepth d. By constructing a treedepth decomposition, the graph is split into blocks (2‑connected components) and a block‑cut tree. The algorithm proceeds in two phases. In Phase I (inside blocks) it handles three cases according to the size of a separator (the “modulator”). If the separator has three or more vertices, each attached component forms a K1,3 “claw”; bounded treedepth forces each claw to be small, and too many claws would force an edge to receive more than one crossing, contradicting 1‑planarity. If the separator has exactly two vertices, the algorithm groups child components by whether they can be drawn with the two separator vertices on the outer face. Those that can are proved “glueable” using Thomassen’s B/W‑free condition and are safely removed. The remaining components are bounded in number by a function of d, otherwise a crossing‑capacity argument fails. Phase II (across blocks) repeats a similar analysis on the block‑cut tree, discarding glueable subtrees and bounding the degree of cut vertices. The result is a bound on the total number of blocks and on the size of each block that depends only on d, yielding a running time of O(2^{2^{O(d)}}·n^{O(1)}).

The second contribution is a kernelization with respect to the feedback‑edge number ℓ. The authors improve the previous O((3ℓ)!) kernel for (topological) 1‑planarity to O(ℓ·8^ℓ) edges, and for geometric 1‑planarity to O(ℓ·27^ℓ) edges. After removing a feedback‑edge set, the remaining graph consists of ℓ edge‑disjoint degree‑2 paths. Paths are classified as “short” or “long”. A global redrawing argument based on Reidemeister moves shows that any two long paths can be redrawn to intersect at most once, allowing the authors to truncate each long path to a constant number of edges while preserving any possible 1‑planar solution. Short paths are kept intact, and the total number of edges after truncation is bounded by ℓ·8^ℓ. For the geometric variant, the same truncation is combined with a check for B‑ and W‑configurations, inflating the constant factor to 27.

The third contribution generalizes the kernelization to geometric k‑planarity (k≥1). Since no straightening characterization exists for k>1, the authors use a different approach: they triangulate the graph with respect to the short paths and embed each long path inside a triangle, guaranteeing that each edge receives at most k crossings. This yields a kernel of size O(2^{O(3ℓ log ℓ)}) edges.

Finally, the paper establishes matching lower bounds. By a novel reduction from Bin Packing, the authors prove that Geometric 1‑Planarity remains NP‑complete even when the input graph has pathwidth at most 15 or feedback‑vertex number at most 48. They also show that bounded‑bandwidth instances stay NP‑complete by replacing each edge with a constant‑size gadget, which only quadratically inflates the bandwidth, and then invoking a known bounded‑bandwidth hardness for topological 1‑planarity. These hardness results demonstrate that the positive algorithmic results are essentially optimal within the usual parameter hierarchy.

In summary, the work delivers (i) an FPT algorithm parameterized by treedepth, (ii) substantially smaller kernels parameterized by feedback‑edge number for both topological and geometric variants, (iii) extensions to geometric k‑planarity, and (iv) strong NP‑hardness proofs for several restrictive structural parameters. The combination of Thomassen’s embedding characterization with refined combinatorial arguments advances our understanding of geometric graph drawing problems from a parameterized‑complexity perspective.