Add walk-distance labelling to source-dual cut

Label each dual face by its distance, within the source-dual cut, from
the first entry's cap down tooth. Regenerate the seed 1 and seed 2 full
walk figures and metadata with the new labelling.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_cuts/experiments/full_walk/full_medial_tire_cut_walk_1.md

@@ -5,13 +5,24 @@
 - deep-embedded edges: 84
 - graph seed: 59
 - deep-embedded minimum degree: 3
+- chosen face: (8, 9, 19)
 - chosen source cap vertex: 24
+- root entry tooth: e2 (apex medial vertex = level-1 edge (19, 8))
 - recognised treads: 11
-- skipped treads: [(0, 'only 0 up teeth')]
+- skipped treads: [((0, 0), 'only 0 up teeth')]
 - removed source-dual edges: 29
 - annular/cap cuts: 12
 - up-apex cuts: 17
 
+- **source-dual cut is a tree: True** (56 dual faces, 55 edges, 1 component(s), acyclic=True)
+
+## Walk-distance labelling of the source-dual cut
+
+Each dual face (vertex of the source-dual cut) is labelled by its distance, within the cut, from the **cap down tooth of the first entry**: the triangular face `(8, 24, 19)` = `{source 24, edge (19, 8)}` (dual node 34).
+
+- maximum distance: 21
+- distance histogram (faces by distance): `{0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 2, 6: 3, 7: 3, 8: 4, 9: 4, 10: 4, 11: 4, 12: 3, 13: 3, 14: 2, 15: 2, 16: 1, 17: 2, 18: 3, 19: 4, 20: 2, 21: 1}`
+
 - dual cut figure: `full_medial_tire_cut_walk_1_dual.png`
 - tire cut grid: `full_medial_tire_cut_walk_1_tires.png`
 - combined PDF: `full_medial_tire_cut_walk_1.pdf`

papers/medial_tire_cuts/experiments/full_walk/full_medial_tire_cut_walk_2.md

@@ -7,7 +7,7 @@
 - deep-embedded minimum degree: 3
 - chosen face: (4, 12, 11)
 - chosen source cap vertex: 20
-- root entry tooth: e3
+- root entry tooth: e3 (apex medial vertex = level-1 edge (11, 4))
 - recognised treads: 3
 - skipped treads: [((0, 0), 'only 0 up teeth')]
 - removed source-dual edges: 20
@@ -16,6 +16,13 @@
 
 - **source-dual cut is a tree: True** (38 dual faces, 37 edges, 1 component(s), acyclic=True)
 
+## Walk-distance labelling of the source-dual cut
+
+Each dual face (vertex of the source-dual cut) is labelled by its distance, within the cut, from the **cap down tooth of the first entry**: the triangular face `(4, 20, 11)` = `{source 20, edge (11, 4)}` (dual node 15).
+
+- maximum distance: 17
+- distance histogram (faces by distance): `{0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 3, 7: 3, 8: 3, 9: 3, 10: 4, 11: 3, 12: 2, 13: 1, 14: 1, 15: 1, 16: 1, 17: 1}`
+
 - dual cut figure: `full_medial_tire_cut_walk_2_dual.png`
 - tire cut grid: `full_medial_tire_cut_walk_2_tires.png`
 - combined PDF: `full_medial_tire_cut_walk_2.pdf`

papers/medial_tire_cuts/experiments/medial_tire_dual_cut_experiment.py

@@ -221,6 +221,44 @@ def dual_face_missing(G, removed):
             for v in G.nodes()}
 
 
+def dual_cut_graph(result):
+    """The source-dual cut itself: the dual graph with every removed (cut) dual
+    edge deleted.  This is the graph the walk distances are read on."""
+    D = result["dual"].copy()
+    removed = result["removed_dual_edges"]
+    D.remove_edges_from([(u, v) for u, v, d in D.edges(data=True)
+                         if d["primal"] in removed])
+    return D
+
+
+def entry_down_tooth_face(result):
+    """Index of the cap down tooth of the first entry: the triangular face
+    ``{source, x, y}`` where ``(x, y)`` is the first entry medial vertex (the
+    root tread's entry up-tooth apex, a level-1 edge that is a down tooth of the
+    source cap).  This dual node roots the walk-distance labelling.  Returns
+    None if no such face exists."""
+    em = result.get("entry_medial_vertex")
+    if em is None:
+        return None
+    target = frozenset((result["source"], em[0], em[1]))
+    for fi, f in enumerate(result["faces"]):
+        if frozenset(f) == target:
+            return fi
+    return None
+
+
+def dual_cut_distances(result):
+    """Distance of every dual face from the first entry's cap down tooth,
+    measured in the source-dual cut.
+
+    Returns ``(dist, root)`` where ``dist`` maps each reachable dual face to its
+    distance and ``root`` is the rooting dual face (or None if absent)."""
+    root = entry_down_tooth_face(result)
+    if root is None:
+        return {}, None
+    return nx.single_source_shortest_path_length(dual_cut_graph(result), root), root
+
+
 # --------------------------------------------------------------------------- #
 # The four chained entry points.
 # --------------------------------------------------------------------------- #
@@ -420,6 +458,7 @@ def _draw_dual_cut_ax(ax, result):
     missing = result["dual_face_missing"]
     source = result["source"]
     entry_medial = result.get("entry_medial_vertex")
+    dist, root = dual_cut_distances(result)
     pos_v = _radial_source_layout(G, source, result["levels"])
 
     def centroid(fi):
@@ -454,9 +493,20 @@ def _draw_dual_cut_ax(ax, result):
                     color="0.80" if cut else "0.25",
                     lw=1.0 if cut else 1.3,
                     linestyle=(0, (2, 2)) if cut else "solid", zorder=1)
+    # dual nodes labelled by walk distance from the first entry's cap down
+    # tooth; the rooting dual face is gold.
     for fi in dual.nodes():
         x, y = pos[fi]
-        ax.plot(x, y, "o", ms=4, color="#3a6ea5", zorder=3)
+        d = dist.get(fi)
+        if fi == root:
+            ax.plot(x, y, "o", ms=12, mfc="#ffd24d", mec="#b8860b",
+                    mew=1.3, zorder=3)
+        else:
+            ax.plot(x, y, "o", ms=10, mfc="white", mec="#3a6ea5",
+                    mew=0.8, zorder=3)
+        ax.text(x, y, str(d) if d is not None else "∞",
+                color="#13335c", fontsize=5.5, fontweight="bold",
+                ha="center", va="center", zorder=4)
     # label each source-graph vertex by its id; the cap source is flagged.
     for v in G.nodes():
         x, y = pos_v[v]
@@ -475,12 +525,15 @@ def _draw_dual_cut_ax(ax, result):
                     ha="left", va="bottom", zorder=7,
                     bbox=dict(boxstyle="circle,pad=0.05", fc="white",
                               ec="#b03030", lw=0.6))
+    root_face = result["faces"][root] if root is not None else None
+    max_dist = max(dist.values()) if dist else 0
     ax.set_title(f"source-dual cut (cap source {source}, entry "
                  f"e{result['entry_edge']} = medial vtx {entry_medial}); "
                  f"gray = edges missing after cuts\n"
-                 f"green star = first entry medial vertex; red numbers = "
+                 f"blue dual-node numbers = distance from gold root "
-                 f"#missing dual edges around each dual face; "
+                 f"(cap down tooth {root_face} of first entry; max {max_dist}); "
-                 f"max {result['max_missing']}", fontsize=9)
+                 f"green star = first entry; red = #missing dual edges per face",
+                 fontsize=9)
     ax.set_aspect("equal")
     ax.axis("off")