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Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...
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Flip Distances Between Graph Orientations.

Oswin Aichholzer1, Jean Cardinal2, Tony Huynh3

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Algorithmica
|February 15, 2021
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Summary
This summary is machine-generated.

Determining the minimum flips between graph orientations is NP-complete for planar graphs, but solvable in polynomial time for specific cases involving sinks and sources.

Keywords:
Flip distanceGraph orientationα-Orientation

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Area of Science:

  • Discrete Mathematics
  • Graph Theory
  • Computational Complexity

Background:

  • Flip graphs model local changes in combinatorial objects, with applications in geometry and combinatorics.
  • Flip distance quantifies the minimum number of local changes (flips) to transform one object into another.
  • Orientations of graphs, specifically k-orientations and orientations with specified cycle properties, are studied.

Purpose of the Study:

  • To investigate the computational complexity of flip distance problems on graph orientations.
  • To analyze the intractability of finding geodesics on combinatorial polytopes related to graph orientations.
  • To explore polynomial-time solvable cases within the broader framework of flip distance computations.

Main Methods:

  • Proving NP-completeness for deciding if the flip distance between two k-orientations of a planar graph is at most two.
  • Relating flip distance problems to finding geodesics on partition and alcoved polytopes.
  • Utilizing the distributive lattice structure of flip graphs for specific cases (sink-source flips).

Main Results:

  • Deciding flip distance for k-orientations of planar graphs is NP-complete, even for perfect matchings.
  • The problem is computationally intractable despite connections to geodesics on combinatorial polytopes.
  • A polynomial-time solution exists for flip distances when flips are restricted to changing sinks to sources or vice-versa.

Conclusions:

  • Flip distance problems on graph orientations exhibit significant computational complexity.
  • The study highlights the intractability of geodesic problems on related polytopes.
  • Restricted flip operations on graph orientations can lead to efficient algorithmic solutions.