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For graphoid automata, these results give polynomial-time recognition without a graph-width restriction, effective boundary composition, and comparison of finite graph relations, and the quadratic boundary bounds are optimal in the worst case.
It is proved that equal closed expressions of pattern width at most $k$ are joined by a derivation in which every step applies an equation in either direction and every intermediate width is at most a computable $B_\Sigma(k)$, independently of graph size.
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