Complexity results in graph reconstruction

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c>=1 of deletion:(1)GI="i"s"o^lVDC"c, GI="i"s"o^lEDC"c, GI= =2, GI="i"s"o^pk-VDC"c and GI="i"s"o^pk-EDC"c. (3)For all k>=2, GI= =2, GI="i"s"o^pk-LED"c. For many of these results, even the c=1 case was not previously known. Similar to the definition of reconstruction numbers vrn"@?(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451-454] and ern"@?(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn"@?(G) and ern"@?(G), and give an example of a family {G"n}"n">="4 of graphs on n vertices for which vrn"@?(G"n)=2 and n>=1, we show that there exists a collection of k graphs on (2^k^-^1+1)n+k vertices with 2^n 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.