Locally grid graphs: classification and Tutte uniqueness

We define a locally grid graph as a graph in which the structure around each vertex is a 3x3 grid @?, the canonical examples being the toroidal grids C"pxC"q. The paper contains two main results. First, we give a complete classification of locally grid graphs, showing that each of them has a natural embedding in the torus or in the Klein bottle. Secondly, as a continuation of the research initiated in (On graphs determined by their Tutte polynomials, Graphs Combin., to appear), we prove that C"pxC"q is uniquely determined by its Tutte polynomial, for p,q>=6.