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Given a polyomino, the "adjacency graph" has one vertex for each tile and an edge connecting tiles which are adjacent (diagonal doesn't count). Is anything known about which graphs can be the adjacency graph of a polyomino? Obviously they must be planar and connected, with no triangles (actually no odd length cycles), and no vertex degree greater than 4. But probably more is required. Has this been studied before?

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    $\begingroup$ I think the polyominoes being discussed are connected unions of cells in the unit square lattice. So they inherit being bipartite from the unit square lattice (just colour it in a chessboard fashion), and so have no odd cycles: in particular no triangles. $\endgroup$ – James Cranch Mar 10 '15 at 12:25
  • $\begingroup$ Certainly more is required: the hexagon fulfils your conditions but is not such a graph. $\endgroup$ – James Cranch Mar 10 '15 at 12:25
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    $\begingroup$ I think these are called "grid graphs". $\endgroup$ – Brendan McKay Mar 10 '15 at 12:42

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