dual_decomposition: reduced-dual definition, verification, and step figures

Add Definition 2.1 (reduced dual) and a remark on cubicity/planarity, plus an
experiment verifying it on the icosahedron/dodecahedron and four figures, one
per construction step.

reduced_dual.py builds G' = dodecahedron (dual of the icosahedron), applies the
construction, and confirms the result is a cubic, planar, simple graph whose
dual is a simple triangulation. Finding: the construction is an n -> n-2
reduction (12 -> 10 here), not n-1, since the single apex v_n collapses one more
vertex than a standard pentagon re-triangulation; the result also re-introduces
degree-3 and degree-4 vertices (degree seq [7,5,5,5,5,5,5,4,4,3]).

draw_reduced_dual_steps.py renders fig_reduced_dual_step1..4.png, embedded as a
2x2 grid after the definition.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

papers/dual_decomposition_minimal_counterexamples/experiments/draw_reduced_dual_steps.py

@@ -0,0 +1,174 @@
+"""Draw the four steps of the reduced-dual construction (Definition 2.1).
+
+Uses the dodecahedron G' = dual of the icosahedron, with F_v the inner pentagon,
+as built in reduced_dual.py. Produces fig_reduced_dual_step{1..4}.png.
+"""
+import os
+import math
+import matplotlib.pyplot as plt
+from matplotlib.patches import Polygon
+from matplotlib.lines import Line2D
+
+from reduced_dual import build_dual, apply_reduction
+
+OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
+
+GRAY = '#9ca3af'
+DARK = '#374151'
+GHOST = '#fca5a5'
+DEG2 = '#f59e0b'
+APEX = '#16a34a'
+CHORD = '#2563eb'
+FACE = '#fef9c3'
+
+
+def draw_edges(ax, G, pos, nodes=None, **kw):
+    for u, v in G.edges():
+        if nodes is not None and (u not in nodes or v not in nodes):
+            continue
+        (x0, y0), (x1, y1) = pos[u], pos[v]
+        ax.plot([x0, x1], [y0, y1], **kw)
+
+
+def draw_nodes(ax, pos, nodes, **kw):
+    xs = [pos[v][0] for v in nodes]
+    ys = [pos[v][1] for v in nodes]
+    ax.scatter(xs, ys, **kw)
+
+
+def face_F_polygon(pos):
+    """The new central face F: decagon alternating b_i, c_i clockwise."""
+    order = []
+    for i in range(5):
+        order += [('b', i), ('c', i)]
+    return [pos[v] for v in order]
+
+
+def base_canvas(title):
+    fig, ax = plt.subplots(figsize=(8.5, 8.5))
+    ax.set_aspect('equal')
+    ax.axis('off')
+    ax.set_title(title, fontsize=12)
+    return fig, ax
+
+
+def main():
+    Gp, pos, Fv = build_dual()
+    res = apply_reduction(Gp, pos, Fv, i=0)
+    Ghat, npos, A = res['Ghat'], res['pos'], res['A']
+    v_n, apex_nbrs, chord = res['v_n'], res['apex_nbrs'], res['chord']
+
+    survivors = [v for v in Gp if v not in Fv]          # b, c, d families
+    surv_set = set(survivors)
+    deg2 = list(A)                                      # the five b_i
+
+    # surviving edges (both endpoints survive) vs deleted edges (touch an a_i)
+    surv_edges = [(u, v) for u, v in Gp.edges()
+                  if u in surv_set and v in surv_set]
+    del_edges = [(u, v) for u, v in Gp.edges()
+                 if u not in surv_set or v not in surv_set]
+
+    def draw_surviving(ax):
+        ax.add_patch(Polygon(face_F_polygon(pos), closed=True,
+                             facecolor=FACE, edgecolor='none', zorder=0))
+        for u, v in surv_edges:
+            (x0, y0), (x1, y1) = pos[u], pos[v]
+            ax.plot([x0, x1], [y0, y1], color=GRAY, lw=1.6, zorder=1)
+        others = [v for v in survivors if v not in deg2]
+        draw_nodes(ax, pos, others, s=120, color=DARK, zorder=3)
+
+    def draw_ghosts(ax):
+        for u, v in del_edges:
+            (x0, y0), (x1, y1) = pos[u], pos[v]
+            ax.plot([x0, x1], [y0, y1], color=GHOST, lw=1.2, ls='--', zorder=1)
+        draw_nodes(ax, pos, Fv, s=120, color='white', edgecolors=GHOST,
+                   linewidths=1.5, zorder=2)
+        for v in Fv:
+            ax.plot(*pos[v], marker='x', color=GHOST, ms=8, zorder=3)
+
+    # ----- Step 1: delete F_v's boundary; five degree-2 vertices on face F -----
+    fig, ax = base_canvas(
+        "Step 1: delete the five dual vertices on $\\partial F_v$.\n"
+        "Their outer neighbours drop to degree 2 (orange) and lie on a new "
+        "face $F$ (shaded).")
+    draw_surviving(ax)
+    draw_ghosts(ax)
+    draw_nodes(ax, pos, deg2, s=260, color=DEG2, edgecolors='black',
+               linewidths=1.0, zorder=4)
+    cx = sum(pos[('a', i)][0] for i in range(5)) / 5
+    cy = sum(pos[('a', i)][1] for i in range(5)) / 5
+    ax.text(cx, cy, '$F$', fontsize=16, ha='center', va='center',
+            color='#a16207', zorder=5)
+    ax.legend(handles=[
+        Line2D([0], [0], marker='x', color=GHOST, lw=0, label='deleted (was $\\partial F_v$)'),
+        Line2D([0], [0], marker='o', color='w', markerfacecolor=DEG2,
+               markeredgecolor='black', label='degree-2 vertex'),
+    ], loc='upper left', fontsize=10)
+    fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step1.png'),
+                dpi=170, bbox_inches='tight'); plt.close(fig)
+
+    # ----- Step 2: order the five degree-2 vertices clockwise as A_0..A_4 -----
+    fig, ax = base_canvas(
+        "Step 2: list the degree-2 vertices clockwise around $F$ as "
+        "$A_0,\\dots,A_4$.")
+    draw_surviving(ax)
+    draw_nodes(ax, pos, deg2, s=300, color=DEG2, edgecolors='black',
+               linewidths=1.0, zorder=4)
+    for k, v in enumerate(A):
+        x, y = pos[v]
+        ax.annotate(f'$A_{k}$', (x, y), textcoords='offset points',
+                    xytext=(0, 0), ha='center', va='center', fontsize=10,
+                    fontweight='bold', color='black', zorder=5)
+        # outward label too
+        ax.annotate(f'$A_{k}$', (x * 1.18, y * 1.18), ha='center', va='center',
+                    fontsize=12, color='#a16207', zorder=5)
+    fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step2.png'),
+                dpi=170, bbox_inches='tight'); plt.close(fig)
+
+    # ----- Step 3: add v_n joined to A_i, A_{i+1}, A_{i+2} -----
+    fig, ax = base_canvas(
+        "Step 3: add a vertex $v_n$ joined to $A_i, A_{i+1}, A_{i+2}$ "
+        "(here $i=0$).")
+    draw_surviving(ax)
+    draw_nodes(ax, pos, deg2, s=300, color=DEG2, edgecolors='black',
+               linewidths=1.0, zorder=4)
+    for k, v in enumerate(A):
+        ax.annotate(f'$A_{k}$', (pos[v][0] * 1.18, pos[v][1] * 1.18),
+                    ha='center', va='center', fontsize=12, color='#a16207', zorder=5)
+    for u in apex_nbrs:
+        (x0, y0), (x1, y1) = npos[v_n], pos[u]
+        ax.plot([x0, x1], [y0, y1], color=APEX, lw=2.4, zorder=5)
+    draw_nodes(ax, npos, [v_n], s=320, color=APEX, marker='s',
+               edgecolors='black', linewidths=1.0, zorder=6)
+    ax.annotate('$v_n$', npos[v_n], textcoords='offset points', xytext=(0, 14),
+                ha='center', fontsize=12, fontweight='bold', color=APEX, zorder=7)
+    fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step3.png'),
+                dpi=170, bbox_inches='tight'); plt.close(fig)
+
+    # ----- Step 4: add chord A_{i+3} A_{i+4}; the reduced dual -----
+    fig, ax = base_canvas(
+        "Step 4: add the edge $A_{i+3} A_{i+4}$. The result $\\widehat{G}'_{v,i}$ "
+        "is again cubic and planar.")
+    draw_surviving(ax)
+    draw_nodes(ax, pos, deg2, s=300, color=DEG2, edgecolors='black',
+               linewidths=1.0, zorder=4)
+    for k, v in enumerate(A):
+        ax.annotate(f'$A_{k}$', (pos[v][0] * 1.18, pos[v][1] * 1.18),
+                    ha='center', va='center', fontsize=12, color='#a16207', zorder=5)
+    for u in apex_nbrs:
+        (x0, y0), (x1, y1) = npos[v_n], pos[u]
+        ax.plot([x0, x1], [y0, y1], color=APEX, lw=2.4, zorder=5)
+    draw_nodes(ax, npos, [v_n], s=320, color=APEX, marker='s',
+               edgecolors='black', linewidths=1.0, zorder=6)
+    ax.annotate('$v_n$', npos[v_n], textcoords='offset points', xytext=(0, 14),
+                ha='center', fontsize=12, fontweight='bold', color=APEX, zorder=7)
+    (x0, y0), (x1, y1) = pos[chord[0]], pos[chord[1]]
+    ax.plot([x0, x1], [y0, y1], color=CHORD, lw=2.8, zorder=5)
+    fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step4.png'),
+                dpi=170, bbox_inches='tight'); plt.close(fig)
+
+    print("wrote fig_reduced_dual_step1..4.png to", OUT_DIR)
+
+
+if __name__ == '__main__':
+    main()

papers/dual_decomposition_minimal_counterexamples/experiments/reduced_dual.py

@@ -0,0 +1,184 @@
+"""Reduced dual: construction and verification.
+
+Test input is the icosahedron G (the unique 5-regular triangulation, n=12).
+Its dual G' is the dodecahedron (a cubic plane graph, 20 vertices). We pick a
+degree-5 vertex v of G -- equivalently a pentagonal face F_v of G' -- and apply
+the reduced-dual construction of Definition 2.1:
+
+  1. delete the 5 dual vertices on the boundary of F_v (and incident edges),
+     leaving 5 degree-2 vertices on a new face F;
+  2. order those 5 vertices clockwise around F as A_0..A_4;
+  3. add a vertex v_n joined to A_i, A_{i+1}, A_{i+2};
+  4. add an edge A_{i+3} A_{i+4}.
+
+We verify the result is again a cubic plane graph, and report the triangulation
+it is the dual of (its face count = the primal vertex count), to see how the
+vertex count changes relative to n.
+
+The dodecahedron is built directly in its concentric "Schlegel" layout with
+F_v the inner pentagon, so the figures (draw_reduced_dual_steps.py) are clean.
+"""
+import math
+import networkx as nx
+
+# ---------------------------------------------------------------------------
+# Build G' = dodecahedron with concentric positions; F_v = inner pentagon.
+# Vertex families a (inner pentagon), b, c, d (outer pentagon), 5 each.
+# Angles increase *clockwise* (90 - 72*i deg) so index order is clockwise.
+# ---------------------------------------------------------------------------
+def build_dual():
+    pos = {}
+    R = {'a': 1.0, 'b': 2.2, 'c': 3.6, 'd': 4.8}
+    for i in range(5):
+        for fam in ('a', 'b'):
+            th = math.radians(90 - 72 * i)
+            pos[(fam, i)] = (R[fam] * math.cos(th), R[fam] * math.sin(th))
+        for fam in ('c', 'd'):
+            th = math.radians(90 - 72 * i - 36)   # offset half a step
+            pos[(fam, i)] = (R[fam] * math.cos(th), R[fam] * math.sin(th))
+
+    Gp = nx.Graph()
+    Gp.add_nodes_from(pos)
+    for i in range(5):
+        Gp.add_edge(('a', i), ('a', (i + 1) % 5))        # inner pentagon
+        Gp.add_edge(('a', i), ('b', i))                  # spokes a-b
+        Gp.add_edge(('b', i), ('c', i))                  # b-c
+        Gp.add_edge(('b', i), ('c', (i - 1) % 5))        # b-c (other side)
+        Gp.add_edge(('c', i), ('d', i))                  # spokes c-d
+        Gp.add_edge(('d', i), ('d', (i + 1) % 5))        # outer pentagon
+
+    Fv_boundary = [('a', i) for i in range(5)]           # inner pentagon
+    return Gp, pos, Fv_boundary
+
+
+# ---------------------------------------------------------------------------
+# Face / dual helpers.
+# ---------------------------------------------------------------------------
+def faces_of(G):
+    """Return the list of faces (each a list of vertices) of a plane graph."""
+    ok, emb = nx.check_planarity(G)
+    assert ok, "graph is not planar"
+    seen, faces = set(), []
+    for u in emb:
+        for v in emb[u]:
+            if (u, v) not in seen:
+                faces.append(emb.traverse_face(u, v, mark_half_edges=seen))
+    return faces
+
+
+def dual_of(G):
+    """Combinatorial dual (all faces, including outer) of a plane graph."""
+    faces = faces_of(G)
+    edge_faces = {}
+    for fi, face in enumerate(faces):
+        for j in range(len(face)):
+            e = frozenset((face[j], face[(j + 1) % len(face)]))
+            edge_faces.setdefault(e, []).append(fi)
+    D = nx.MultiGraph()
+    D.add_nodes_from(range(len(faces)))
+    for e, fs in edge_faces.items():
+        if len(fs) == 2:
+            D.add_edge(fs[0], fs[1])
+        elif len(fs) == 1:                # shouldn't happen for 2-connected G
+            pass
+    return D, faces
+
+
+# ---------------------------------------------------------------------------
+# The reduced-dual construction.
+# ---------------------------------------------------------------------------
+def clockwise_order(verts, pos):
+    """Order verts clockwise around their centroid, starting from the topmost."""
+    cx = sum(pos[v][0] for v in verts) / len(verts)
+    cy = sum(pos[v][1] for v in verts) / len(verts)
+    ang = {v: math.atan2(pos[v][1] - cy, pos[v][0] - cx) for v in verts}
+    ccw = sorted(verts, key=lambda v: ang[v])            # counterclockwise
+    cw = list(reversed(ccw))                             # clockwise
+    start = max(range(len(cw)), key=lambda k: pos[cw[k]][1])   # topmost first
+    return cw[start:] + cw[:start]
+
+
+def apply_reduction(Gp, pos, Fv_boundary, i=0):
+    """Apply Definition 2.1 and return a dict capturing each stage."""
+    Ghat = Gp.copy()
+    npos = dict(pos)
+
+    # (1) delete the 5 boundary dual vertices of F_v
+    Ghat.remove_nodes_from(Fv_boundary)
+    deg2 = [v for v in Ghat if Ghat.degree(v) == 2]
+    assert len(deg2) == 5, f"expected 5 degree-2 vertices, got {len(deg2)}"
+
+    # (2) order them clockwise around the new face F
+    A = clockwise_order(deg2, pos)
+
+    # (3) new vertex v_n joined to A_i, A_{i+1}, A_{i+2}
+    apex_nbrs = [A[(i + k) % 5] for k in range(3)]
+    ax = sum(npos[v][0] for v in apex_nbrs) / 3
+    ay = sum(npos[v][1] for v in apex_nbrs) / 3
+    v_n = 'v_n'
+    npos[v_n] = (ax * 0.55, ay * 0.55)                   # pull toward the 3 nbrs
+    Ghat.add_node(v_n)
+    for u in apex_nbrs:
+        Ghat.add_edge(v_n, u)
+
+    # (4) chord between the remaining two
+    chord = (A[(i + 3) % 5], A[(i + 4) % 5])
+    Ghat.add_edge(*chord)
+
+    return {
+        'Ghat': Ghat, 'pos': npos, 'A': A, 'v_n': v_n,
+        'apex_nbrs': apex_nbrs, 'chord': chord,
+        'deleted': list(Fv_boundary),
+    }
+
+
+def main():
+    Gp, pos, Fv = build_dual()
+
+    # --- verify G' is the dodecahedron = dual of the icosahedron ---
+    assert nx.check_planarity(Gp)[0]
+    assert all(d == 3 for _, d in Gp.degree()), "G' not cubic"
+    assert nx.is_isomorphic(Gp, nx.dodecahedral_graph()), "G' is not dodecahedron"
+    Dico, _ = dual_of(Gp)
+    Dico = nx.Graph(Dico)
+    print(f"G  (icosahedron) : dual of G' has {Dico.number_of_nodes()} vertices, "
+          f"degrees {sorted({d for _, d in Dico.degree()})}")
+    print(f"G' (dodecahedron): {Gp.number_of_nodes()} vertices, "
+          f"{Gp.number_of_edges()} edges, "
+          f"{len(faces_of(Gp))} faces; cubic={all(d==3 for _,d in Gp.degree())}")
+
+    # --- apply the reduced-dual construction ---
+    res = apply_reduction(Gp, pos, Fv, i=0)
+    Ghat = res['Ghat']
+    cubic = all(d == 3 for _, d in Ghat.degree())
+    planar = nx.check_planarity(Ghat)[0]
+    ghat_simple = (nx.number_of_selfloops(Ghat) == 0)   # Graph: no parallels
+    nfaces = len(faces_of(Ghat))
+    print()
+    print(f"reduced dual  G^_v,i : {Ghat.number_of_nodes()} vertices, "
+          f"{Ghat.number_of_edges()} edges, {nfaces} faces")
+    print(f"  cubic   : {cubic}")
+    print(f"  planar  : {planar}")
+    print(f"  simple  : {ghat_simple}")
+
+    # --- the triangulation it is dual to ---
+    Dred_multi, _ = dual_of(Ghat)
+    Dred = nx.Graph(Dred_multi)
+    dred_simple = (Dred.number_of_edges() == Dred_multi.number_of_edges())
+    is_tri = all(len(f) == 3 for f in faces_of(Dred)) if planar else None
+    print()
+    print(f"dual of reduced dual : {Dred.number_of_nodes()} vertices "
+          f"(= faces of G^), degree seq "
+          f"{sorted((d for _, d in Dred.degree()), reverse=True)}")
+    print(f"  is a triangulation : {is_tri}")
+    print(f"  simple             : {dred_simple}")
+
+    n = Dico.number_of_nodes()
+    print()
+    print(f"VERTEX COUNT: G has n = {n}; reduced triangulation has "
+          f"{Dred.number_of_nodes()} (change = "
+          f"{Dred.number_of_nodes() - n}).")
+
+
+if __name__ == '__main__':
+    main()

papers/dual_decomposition_minimal_counterexamples/paper.aux

@@ -8,4 +8,8 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\gdef \@abspage@last{2}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The reduced dual}}{2}{}\protected@file@percent }
+\newlabel{def:reduced-dual}{{2.1}{2}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 2.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots  ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{3}{}\protected@file@percent }
+\newlabel{fig:reduced-dual-steps}{{1}{3}}
+\gdef \@abspage@last{3}

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papers/dual_decomposition_minimal_counterexamples/paper.tex

@@ -115,4 +115,63 @@ $G$, a contradiction.
 Hence $\delta(G) \ge 5$.
 \end{proof}
 
+\section{The reduced dual}
+
+Write $G'$ for the dual of $G$: since $G$ is a triangulation, $G'$ is a cubic
+plane graph in which each vertex of $G$ corresponds to a face of $G'$, each face
+of $G$ to a vertex of $G'$, and each edge to a dual edge. A vertex of $G$ of
+degree $k$ corresponds to a $k$-gonal face of $G'$.
+
+By Lemma~\ref{lem:mindeg}, $\delta(G) \ge 5$, and Euler's formula gives
+$\sum_{u \in V(G)}(6 - \deg u) = 12$, so $G$ has a vertex of degree exactly $5$
+(indeed at least twelve). Fix such a vertex $v$. Its dual face $F_v$ is a
+pentagon, bounded by the five dual vertices corresponding to the five faces of
+$G$ incident to $v$.
+
+\begin{definition}[Reduced dual]
+\label{def:reduced-dual}
+Let $v$ be a degree-$5$ vertex of $G$ with pentagonal dual face $F_v$, and fix an
+index $i \in \{0,1,2,3,4\}$. The \emph{reduced dual} $\widehat{G}'_{v,i}$ is the
+plane graph obtained from $G'$ as follows.
+\begin{enumerate}
+  \item Delete the five dual vertices on the boundary of $F_v$, together with all
+        edges incident to them. Each deleted vertex is cubic, with two edges on
+        $\partial F_v$ and one edge leaving $F_v$; deleting the five boundary
+        vertices therefore removes the five external edges as well, dropping their
+        five outer endpoints from degree $3$ to degree $2$. These five degree-$2$
+        vertices lie on the boundary of a single face $F$ of the resulting graph.
+  \item List the five degree-$2$ vertices in clockwise order around $F$ as
+        $A = (A_0, A_1, A_2, A_3, A_4)$.
+  \item Add a new vertex $v_n$ and join it to $A_i$, $A_{i+1}$, and $A_{i+2}$
+        (indices mod $5$) by three new edges.
+  \item Add a new edge between $A_{i+3}$ and $A_{i+4}$ (indices mod $5$).
+\end{enumerate}
+\end{definition}
+
+\begin{remark}
+Steps (3) and (4) restore cubicity: $A_i, A_{i+1}, A_{i+2}$ each gain one edge to
+$v_n$ and $A_{i+3}, A_{i+4}$ each gain the new edge, so all five return to degree
+$3$, and $v_n$ has degree $3$. Since $A_i,\dots,A_{i+2}$ and $A_{i+3}, A_{i+4}$
+are each consecutive along $\partial F$, the new vertex and edge can be drawn
+inside $F$ without crossings, so $\widehat{G}'_{v,i}$ is again a cubic plane
+graph. The construction depends on the choice of $i$ up to the rotational
+symmetry of $A$.
+\end{remark}
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step1.png}\hfill
+\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step2.png}\\[0.5em]
+\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step3.png}\hfill
+\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step4.png}
+\caption{The four steps of Definition~\ref{def:reduced-dual}, illustrated on
+$G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner
+pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$,
+leaving five degree-$2$ vertices on a new face $F$. Top right: order them
+clockwise as $A_0,\dots,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$.
+Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph
+$\widehat{G}'_{v,0}$.}
+\label{fig:reduced-dual-steps}
+\end{figure}
+
 \end{document}