A124003 - OEIS

%I #9 Aug 09 2020 09:19:44

%S 3,2,9,7,19,8,8,56,90,2,16,496,520,12,0,266,8308,3584,284,4,0,45186,

%T 199247,28781,1434,20,0,0,6054148,5637886,301530,10686,914,4,0,0

%N 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 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 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 B. McMullen, <a href="https://www.semanticscholar.org/paper/Graph-reconstruction-numbers-McMullen/09d3f018cdc3feedd30d5ed32ccca9344cc180c4">Graph reconstruction numbers</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Reconstruction_conjecture">Reconstruction conjecture</a>.

%e Triangle begins

%e         3

%e         2       9

%e         7      19      8

%e         8      56     90     2

%e        16     496    520    12   0

%e       266    8308   3584   284   4 0

%e     45186  199247  28781  1434  20 0 0

%e   6054148 5637886 301530 10686 914 4 0 0

%Y Cf. A124002, A000088, A006652-A006655.

%K hard,more,nice,nonn,tabl

%O 3,1

%A _Martin Fuller_, Dec 08 2006