  We introduce a notion of "simulation" for labelled graphs, in which edges of
the simulated graph are realized by regular expressions in the simulating
graph, and prove that the tiling problem (aka "domino problem") for the
simulating graph is at least as difficult as that for the simulated graph.
  We apply this to the Cayley graph of the "lamplighter group" $L=\mathbb
Z/2\wr\mathbb Z$, and more generally to "Diestel-Leader graphs". We prove that
these graphs simulate the plane, and thus deduce that the seeded tiling problem
is unsolvable on the group $L$.
  We note that $L$ does not contain any plane in its Cayley graph, so our
undecidability criterion by simulation covers cases not covered by Jeandel's
criterion based on translation-like action of a product of finitely generated
infinite groups.
  Our approach to tiling problems is strongly based on categorical
constructions in graph theory.
