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A124003 Triangle T(n,k) of the number of unlabeled graphs on n nodes with universal reconstruction number k, 3<=k<=n. URN(G) is the minimum size for which all multisubsets of vertex-deleted subgraphs of G can uniquely reconstruct G up to isomorphism.

%I A124003 #9 Aug 09 2020 09:19:44
%S A124003 3,2,9,7,19,8,8,56,90,2,16,496,520,12,0,266,8308,3584,284,4,0,45186,
%T A124003 199247,28781,1434,20,0,0,6054148,5637886,301530,10686,914,4,0,0
%N A124003 Triangle T(n,k) of the number of unlabeled graphs on n nodes with universal reconstruction number k, 3<=k<=n. URN(G) is the minimum size for which all multisubsets of vertex-deleted subgraphs of G can uniquely reconstruct G up to isomorphism.
%C A124003 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 URN and so sum(k=3,n,T(n,k))==A000088(n) for all n>=3.
%H A124003 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 A124003 B. McMullen, <a href="https://www.semanticscholar.org/paper/Graph-reconstruction-numbers-McMullen/09d3f018cdc3feedd30d5ed32ccca9344cc180c4">Graph reconstruction numbers</a>.
%H A124003 Wikipedia, <a href="http://en.wikipedia.org/wiki/Reconstruction_conjecture">Reconstruction conjecture</a>.
%e A124003 Triangle begins
%e A124003         3
%e A124003         2       9
%e A124003         7      19      8
%e A124003         8      56     90     2
%e A124003        16     496    520    12   0
%e A124003       266    8308   3584   284   4 0
%e A124003     45186  199247  28781  1434  20 0 0
%e A124003   6054148 5637886 301530 10686 914 4 0 0
%Y A124003 Cf. A124002, A000088, A006652-A006655.
%K A124003 hard,more,nice,nonn,tabl
%O A124003 3,1
%A A124003 _Martin Fuller_, Dec 08 2006