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A124002 Triangle T(n,k) of the number of unlabeled graphs on n nodes with existential reconstruction number k, 3<=k<=n. ERN(G) is the minimum number of vertex-deleted subgraphs of G required to uniquely reconstruct G up to isomorphism.

%I A124002 #9 Aug 09 2020 09:19:50
%S A124002 4,8,3,34,0,0,150,4,2,0,1044,0,0,0,0,12334,8,2,2,0,0,274666,0,2,0,0,0,
%T A124002 0,12005156,6,4,0,2,0,0,0
%N A124002 Triangle T(n,k) of the number of unlabeled graphs on n nodes with existential reconstruction number k, 3<=k<=n. ERN(G) is the minimum number of vertex-deleted subgraphs of G required to uniquely reconstruct G up to isomorphism.
%C A124002 The (vertex) Reconstruction Conjecture, due to Kelly and Ulam, states that every graph with three or more vertices is reconstructible up to isomorphism given the multiset of vertex deleted subgraphs. Equivalently, every graph has an ERN and so sum(k=3,n,T(n,k))==A000088(n) for all n>=3.
%H A124002 P. J. Kelly, <a href="https://projecteuclid.org/euclid.pjm/1103043674">A congruence theorem for trees</a>, Pacific J. Math., 7 (1957), 961-968.
%H A124002 B. McMullen, <a href="https://www.semanticscholar.org/paper/Graph-reconstruction-numbers-McMullen/09d3f018cdc3feedd30d5ed32ccca9344cc180c4">Graph reconstruction numbers</a>.
%H A124002 Wikipedia, <a href="http://en.wikipedia.org/wiki/Reconstruction_conjecture">Reconstruction conjecture</a>.
%e A124002 Triangle begins
%e A124002          4
%e A124002          8, 3
%e A124002         34, 0, 0
%e A124002        150, 4, 2, 0
%e A124002       1044, 0, 0, 0, 0
%e A124002      12334, 8, 2, 2, 0, 0
%e A124002     274666, 0, 2, 0, 0, 0, 0
%e A124002   12005156, 6, 4, 0, 2, 0, 0, 0
%Y A124002 Cf. A124003, A000088, A006652-A006655.
%K A124002 hard,more,nice,nonn,tabl
%O A124002 3,1
%A A124002 _Martin Fuller_, Dec 08 2006