Complexity results in graph reconstruction

Abstract

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 ≥ 1, GI ≡isol VDCc, GI =isol EDCc, GI ≤ml LVDc, and GI ≡isoP LEDc. 2. For all c ≥ 1 and k ≥ 2, GI ≡isoP k-VDCc and GI ≡isoP k-EDCc. 3. For all c ≥ 1 and k ≥ 2, GI ≤ml k-LVDc. In particular, for all c ≥ 1, GI ≡isoP 2-LVDc. 4. For all c ≥ 1 and k ≥ 2, GI ≡isoP k-LEDc. For many of these, even the c = 1 cases were not known. Similar to the definition of reconstruction numbers urn∃(G) [10] and ern∃(G) (see p. 120 of [17]), we introduce two new graph parameters, urn∀(G) and ern∀(G), and give an example of a family {Gn} n∼4 of graphs on n vertices for which vrn∃(Gn) < urn∀(Gn). For every k ≥ 2 and n ≥ 1, we show there exists a collection of k graphs on (2k-1 + 1)n+ k vertices with 2n 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. © Springer-Verlag 2004.