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 (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 l VDC c , GI ≡ iso l EDC c , GI ⩽ m l LVD c , and GI ≡ iso p LED c . (2) For all k ⩾ 2 , GI ≡ iso p k - VDC c and GI ≡ iso p k - EDC c . (3) For all k ⩾ 2 , GI ⩽ m l k - LVD c . (4) GI ≡ iso p 2 - LVD c . (5) For all k ⩾ 2 , GI ≡ iso p k - 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 ) < vrn ∀ ( G n ) . For every k ⩾ 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.