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\citation{bauerfeld-nested-tire-duals}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname  \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm  {out}}$ a $6$-cycle on vertices $0,\dots  ,5$ (blue), inner boundary $B_{\mathrm  {in}}$ a $4$-cycle on vertices $6,\dots  ,9$ (red), inner outerplanar graph $O = B_{\mathrm  {in}} \cup \{7\text  {--}9\}$ (with one chord, orange), and $E_{\mathrm  {ann}}$ (grey) tiling the annulus between $B_{\mathrm  {out}}$ and $B_{\mathrm  {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent }
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\citation{bauerfeld-depth}
\citation{bauerfeld-depth}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm  {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{8}{}\protected@file@percent }
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\citation{bauerfeld-nested-tire-duals}
\citation{bauerfeld-nested-tire-duals}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm  {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text  {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph  {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong  \Theta (1, 7, 7)$ is outerplanar.}}{9}{}\protected@file@percent }
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\citation{tait-original}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Seam realizability for a small example. $(a)$ A stacked-ring triangulation $G_1$ with single-vertex source $S_1 = \{0\}$ and concentric levels $L_1, L_2, L_3$; its tree of tire treads is the chain $T_0 \to T_1 \to T_2$ with $O^{(T_d)} = G_1[L_{d+1}]$ a $3$-cycle on each tread. $(b)$ The apex-removal seam construction $H_5 = (G_1 \setminus S_1) \cup A_5$, re-embedded so that the former fan-face around $S_1$ becomes the outer face (with $L_1$ now the outermost $G_1$-derived ring and $L_3$ innermost), and with an annular triangulation $A_5$ (orange) attaching to a fresh $5$-cycle $\partial H_5$ (red). Vertex labels show $\mathrm  {BFS}_{\partial H_5}$ levels in $H_5$: they agree with $\ell _{G_1}$ on $V(G_1) \setminus \{S_1\}$, so $\mathcal  {T}(H_5, \partial H_5)$ is iso (combinatorial, $O$-preserved) to $\mathcal  {T}(G_1, S_1)$.}}{13}{}\protected@file@percent }
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\bibcite{tait-original}{1}
\bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tire-duals}{3}
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