didericis
bb144f069e
- Define bridge switch (E/O switch whose new same-parity edge is a bridge in its parity subgraph) and bridge-derived level graph in the paper. Note that bridge switches preserve bipartite parity subgraphs, so every bridge-derived level graph is automatically valid. - Discover the E/O-switch relation is directed (irreversible when a switch produces a cross-parity edge); T*_9 reaches an ELG forward but no ELG reaches it, explaining why it is not derived. This rules out a simple switch-invariant characterization. - Bridge orbits are far smaller than full E/O orbits (~10^4 vs ~10^8 for some labellings), making exhaustive search feasible. Each of the 4 open duals has ~150 valid parity partitions; exhaustive bridge-orbit search per partition can decide bridge-derivability conclusively. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
54 lines
5.6 KiB
TeX
54 lines
5.6 KiB
TeX
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\newlabel{fig:parity-subgraph}{{4}{3}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.4}{}}
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