The graph reconstruction number
Journal of Graph Theory
The multiple sequence alignment problem in biology
SIAM Journal on Applied Mathematics
Graph isomorphism is in the low hierarchy
Journal of Computer and System Sciences
Restricted Turing reducibilities and the structure of the polynomial time hierarchy
Restricted Turing reducibilities and the structure of the polynomial time hierarchy
The ally-reconstruction number of a tree with five or more vertices is three
Journal of Graph Theory
Journal of the ACM (JACM)
The graph isomorphism problem: its structural complexity
The graph isomorphism problem: its structural complexity
On the complexity of graph reconstruction
Mathematical Systems Theory
Threshold Computation and Cryptographic Security
SIAM Journal on Computing
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Reconstruction of interval graphs
Theoretical Computer Science
Reconstruction algorithm for permutation graphs
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
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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.