Existential theory of the reals → Geometric thickness of simple graphs Complexity open

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Clone https://github.com/reduction-zoo/simple-graph-geometric-thickness to investigate Existential theory of the reals → Geometric thickness of simple graphs. Read the fixed question and state. Use preparation notes, the test corpus, and verifier if present; otherwise complete Prepare before a reduction attempt. Follow the repository instructions and your AutoResearch pipeline.

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Why this question matters

Real-algebraic hardness would explain geometric recognition difficulty beyond ordinary combinatorial NP-hardness.

Real-algebraic hardness in formulations using multigraphs or extra geometric information does not settle geometric thickness of simple graphs alone.

The reduction to construct

The source asks whether a finite existential formula of polynomial equalities and inequalities over the reals is satisfiable, using binary integer coefficients. Recovery must respect the campaign’s finite algebraic witness representation; an unsatisfiable instance returns NO-SOLUTION.

Given a simple graph and layer budget k, find a straight-line drawing and an edge partition into at most k crossing-free layers, or report NO-SOLUTION. Coordinates require a finite real-algebraic encoding.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is ∃R-hardness for the stated target problem.

A simple graph must enforce geometric incidences without the parallel edges or extra geometric data used by other formulations.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-15. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.