# Full vs Reduced Medial Tire Findings

Question: do Definition 3.1 (full medial tire graph) and Definition 3.2
(reduced medial tire graph) differ?

## Experiment

Script:

```bash
python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
```

The script compares two models.

- Ambient tread-face model: medial edges are contributed by annular
  triangular faces of the tire tread inside the ambient triangulation.
- Standalone tire-with-boundary-faces model: the outer and inner
  boundary walks are also treated as faces, as in the older drawing
  script.

## Random Sweep

Command:

```bash
python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
```

Result:

```text
ambient tread-face model
  cases checked: 7200
  cases where full != reduced: 0
  removed-edge reasons: {}

standalone tire-with-boundary-faces model
  cases checked: 7200
  cases where full != reduced: 7200
  removed-edge reasons: {'inner_boundary': 39600, 'outer_boundary': 39600}
  first difference:
    m=3 k=3 requested_chords=0 seed=0
    path=IOOOII chords=[]
    full_edges=24 reduced_edges=18
    removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
```

## Exhaustive Small Sweep

Command:

```bash
python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py --exhaustive --max-cycle 5 --max-chords 2
```

Result:

```text
exhaustive ambient tread-face model
  cases checked: 5578
  cases where full != reduced: 0

exhaustive standalone tire-with-boundary-faces model
  cases checked: 5578
  cases where full != reduced: 5578
  first difference:
    m=3 k=3 chords=() path=OOOIII
    full_edges=24 reduced_edges=18
    removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
```

## Interpretation

For the intended ambient-triangulation definition, the experiments
support the suspicion that Definition 3.1 and Definition 3.2 coincide:
same-boundary medial edges do not arise from annular triangular tread
faces, and inner chords are not incident to tread triangles.

They differ only in the standalone tire-with-boundary-faces model,
where the artificial outer and inner boundary faces create medial edges
between consecutive boundary edges.
