[Paper Review] Picking Planar Edges; or, Drawing a Graph with a Planar Subgraph

Marcus Schaefer|arXiv (Cornell University)|Nov 26, 2013

Computational Geometry and Mesh Generation10 references4 citations

TL;DR

This paper introduces a polynomial-time algorithm for determining whether a graph can be drawn such that a specified subset of edges is crossing-free, using a Hanani-Tutte-style approach over GF(2). It proves that the partial planarity problem is solvable in polynomial time, while the geometric variant—requiring straight-line drawings with at most one crossing per edge in the subset—is as hard as the existential theory of the reals.

ABSTRACT

Given a graph $G$ and a subset $F \subseteq E(G)$ of its edges, is there a drawing of $G$ in which all edges of $F$ are free of crossings? We show that this question can be solved in polynomial time using a Hanani-Tutte style approach. If we require the drawing of $G$ to be straight-line, and allow at most one crossing along each edge in $F$, the problem turns out to be as hard as the existential theory of the real numbers.

Motivation & Objective

• To determine the computational complexity of drawing a graph such that a given subset of edges is crossing-free.
• To investigate whether the partial planarity problem—where only a subset of edges must be planar—can be solved efficiently.
• To explore the boundary between tractable and intractable variants of planarity constraints in graph drawing.
• To compare the complexity of the poly-line and straight-line versions of the partial planarity problem.
• To examine connections with related problems such as partially embedded planarity and simultaneous embeddings with fixed edges.

Proposed method

• Formulate the problem as a system of linear equations over GF(2), modeling edge crossing parities using (e,v)-moves.
• Define a system P(D,F) that enforces even crossing parity between independent edges, where one edge is in the specified subset F.
• Use the Hanani-Tutte characterization: a graph admits a drawing with F crossing-free if and only if P(D,F) is solvable over GF(2).
• Leverage the fact that solvability of linear systems over finite fields can be tested in polynomial time.
• Apply topological and combinatorial arguments to show that solvability implies the existence of a valid drawing.
• Use pseudoline arrangements and stretchability arguments to analyze the geometric variant and its complexity.

Experimental results

Research questions

• RQ1 Can the partial planarity problem—drawing a graph with a specified subset of edges free of crossings—be solved in polynomial time?
• RQ2 Is the geometric variant of the partial planarity problem, requiring straight-line drawings, computationally harder than the poly-line variant?
• RQ3 What is the computational complexity of the geometric partial planarity problem when each edge in the subset is allowed at most one crossing?
• RQ4 Does the Hanani-Tutte approach extend to 1-planar graphs with geometric constraints?
• RQ5 Can traditional data structures like PQ-trees or SPQR-trees be used to solve the partial planarity problem efficiently?

Key findings

• The partial planarity problem—drawing a graph such that a specified subset of edges is crossing-free—can be solved in polynomial time using a Hanani-Tutte-style linear system over GF(2).
• The solvability of the system P(D,F) over GF(2) is both necessary and sufficient for the existence of a drawing where all edges in F are crossing-free.
• The geometric variant of the problem, requiring straight-line drawings with at most one crossing per edge in F, is as hard as the existential theory of the reals (∃R).
• Geometric 1-planarity is NP-complete, but the general geometric partial planarity problem is likely ∃R-hard, based on area lower bounds and structural complexity.
• The problem is equivalent to a special case of simultaneous embedding with fixed edges (the sunflower case), suggesting potential algorithmic connections to that framework.
• The paper leaves open whether geometric partial planarity is ∃R-hard, though strong evidence points toward this complexity.