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We present results on the structure of graphs with polynomial growth. For certain classes of these graphs we prove that they contain subgraphs which are similar to the lattice graph L d . These results are then applied to investigate isoperimetric properties of certain classes of graphs with polynomial growth. In addition these structural results can also be applied to percolation problems.doi:10.1016/s0012-365x(97)00194-5 fatcat:x2gwtbtgx5cbxldaiqvynjckjm