Treewidth t+1 graphs are not first-order transducible from treewidth t graphs, with analogous separations for Hadwiger number and for treewidth 4 graphs from planar graphs.
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First-order transducibility among classes of sparse graphs
Treewidth t+1 graphs are not first-order transducible from treewidth t graphs, with analogous separations for Hadwiger number and for treewidth 4 graphs from planar graphs.