[Paper Review] A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth

Franz J. Brandenburg, Dobler, Alexander+3 more|arXiv (Cornell University)|Sep 11, 2015

Computational Geometry and Mesh Generation27 references16 citations

TL;DR

This paper establishes a one-to-one correspondence between 1-planar graphs and 4-map graphs (with and without holes), proving that 1-planarity can be recognized in polynomial time for crossing-augmented, fully triangulated, and maximal 1-planar graphs—with time complexities of O(n^120), O(n^3), and O(n^5), respectively. The key contribution is a novel characterization linking 1-planar graphs to map graphs and efficient recognition algorithms under structural constraints.

ABSTRACT

A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets 𝒮 of crossing types gives a recognition problem: does a given graph admit an 𝒮-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in 𝒮? We show that there is a set 𝒮_bad with three crossing types and the following properties: - If 𝒮 contains no crossing type from 𝒮_bad, then the recognition of graphs that admit an 𝒮-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. - If 𝒮 contains any crossing type from 𝒮_bad, then it is NP-hard to decide whether a graph has an 𝒮-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.

Motivation & Objective

• To establish a one-to-one correspondence between 1-planar graphs and 4-map graphs, including hole-free variants.
• To investigate the recognition complexity of 1-planar graphs under structural constraints such as maximality and triangulation.
• To extend existing recognition algorithms for 1-planar rotation systems to crossing-augmented 1-planar graphs.
• To resolve open problems regarding the recognition of 1-planar and map graphs, particularly concerning polynomial-time solvability.
• To determine whether maximal or optimal 1-planar graphs have unique embeddings up to weak isomorphism.

Proposed method

• Characterize 4-map graphs as crossing-augmented 1-planar graphs and hole-free 4-map graphs as fully triangulated 1-planar graphs.
• Leverage the recognition algorithm of Chen et al. [18] for hole-free 4-map graphs to derive time bounds for fully triangulated and maximal 1-planar graphs.
• Adapt the linear-time 1-planar rotation system test of Eades et al. [23] to crossing-augmented 1-planar graphs via a bridge-based walk detection mechanism.
• Use SPQR-tree decomposition to test planar-maximality and maximality in linear time for outer 1-planar and related classes.
• Apply the SPQR-tree structure to verify properties like outer 1-planarity, crossing-augmentation, and weak triangulation.
• Reduce the problem of planar-maximality in 1-planar graphs to checking components at separation pairs via SPQR decomposition.

Experimental results

Research questions

• RQ1 Can 1-planar graphs be characterized as 4-map graphs or hole-free 4-map graphs?
• RQ2 Is the recognition of 1-planar graphs polynomial-time solvable under structural constraints like crossing-augmentation, full triangulation, or maximality?
• RQ3 Can the 1-planar rotation system test of Eades et al. be generalized to crossing-augmented 1-planar graphs?
• RQ4 Do maximal or optimal 1-planar graphs have unique embeddings up to weak isomorphism?
• RQ5 Does the recognition problem for generalized map graphs (e.g., with nested regions) remain polynomial-time solvable?

Key findings

• A one-to-one correspondence is established between 1-planar graphs and 4-map graphs, including both general and hole-free variants.
• Fully triangulated 1-planar graphs are equivalent to hole-free 4-map graphs, and their recognition can be performed in O(n^3) time.
• Maximal 1-planar graphs can be recognized in O(n^5) time using the algorithm of Chen et al. [18].
• Crossing-augmented 1-planar graphs can be recognized in O(n^120) time, matching the complexity of the hole-free 4-map graph recognition algorithm.
• A linear-time algorithm is developed to test 1-planarity of rotation systems for crossing-augmented 1-planar graphs.
• The recognition of outer 1-planar graphs and related properties (e.g., maximal, optimal) is solvable in linear time using SPQR-tree decomposition.