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. We show that the problems are rather closely related for all amounts c of deletion: 1. For all c greater than or equal to 1, GI equivalent to (l)(iso) VDCc, GI equivalent to (l)(iso) EDCc, GI less than or equal to (l)(m) LVDc, and GI equivalent to (p)(iso) LEDc. 2. For all c greater than or equal to 1 and k greater than or equal to 2, GI equivalent to (p)(iso) k-VDCc and GI equivalent to (p)(iso) k-EDCc. 3. For all c greater than or equal to 1 and k greater than or equal to 2, GI less than or equal to (l)(m) k-LVDc. In particular, for all c greater than or equal to 1, GI equivalent to (p)(iso) 2-LVDc. 4. For all c greater than or equal to 1 and k greater than or equal to 2, GI equivalent to (p)(iso) k-LEDc. For many of these, even the c = 1 cases were not known. Similar to the definition of reconstruction numbers vrn(There Exists)(G) [10] and ern(There Exists)(G) (see p. 120 of [17]), we introduce two new graph parameters, vrn(For All)(G) and ern(For All)(G), and give an example of a family {G(n)}(n greater than or equal to 4) of graphs on n vertices for which vrn(There Exists)(G(n)) < vrn(For All)(G). For every k greater than or equal to 2 and n greater than or equal to 1, we show 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.