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 ≡iso VDCc, GI ≡iso EDCc, GI ≤m LVDc, and GI ≡iso LEDc. 2. For all k ≥ 2, GI ≡iso k-VDCc and GI ≡iso k-EDCc. 3. For all k ≥ 2, GI ≤m k-LVDc. 4. GI ≡iso 2-LVDc. 5. For all k ≥ 2, GI ≡iso k-LEDc. For many of these results, even the c = 1 case was not previously known. Similar to the definition of reconstruction numbers vrn∃(G) [HP85] and ern∃(G) (see p. 120 of [LS03]), we introduce two new graph parameters, vrn∀(G) and ern∀(G), and give an example of a family {Gn}n≥4 of graphs on n vertices for which vrn∃(Gn) < vrn∀(Gn). For every k ≥ 2 and n ≥ 1, we show that there exists a collection of k graphs on (2k−1 + 1)n + k vertices with 2 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.