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Summary
This summary is machine-generated.

Deciding if an edge can be inserted into a simple graph drawing is NP-complete, even for pseudocircular drawings. However, determining if a pseudosegment can extend to a pseudocircle within an arrangement is possible in polynomial time.

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Area of Science:

  • Graph theory
  • Computational geometry
  • Combinatorial geometry

Background:

  • Simple drawings of graphs model edge intersections.
  • Levi's Enlargement Lemma states that edges can always be inserted into rectilinear or pseudolinear drawings.
  • Pseudocircles offer a more general framework for graph embeddings.

Purpose of the Study:

  • Investigate the complexity of inserting edges into simple graph drawings.
  • Explore the conditions under which a pseudosegment can be extended to a pseudocircle in an arrangement.

Main Methods:

  • NP-completeness proofs for edge insertion problems.
  • Polynomial-time algorithms for pseudocircle extension problems.
  • Analysis of graph drawing properties and pseudolinear/pseudocircular arrangements.

Main Results:

  • It is NP-complete to determine if an edge can be inserted into a simple graph drawing, even for pseudocircular drawings.
  • A polynomial-time algorithm is presented to check if a pseudosegment can be extended to a pseudocircle within a given arrangement.

Conclusions:

  • The problem of edge insertion into simple graph drawings is computationally hard.
  • Efficient algorithms exist for specific pseudocircle-related problems, suggesting tractability in certain geometric settings.