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The vertex and edge graph reconstruction numbers of small graphs

First posed in 1942 by Kelly and Ulam, the Graph Reconstruction Conjecture is one of the major open problem in graph theory. While the Graph Reconstruction Conjecture remains open, it has spawned a number of related questions. In the classical vertex graph reconstruction number problem a vertex is deleted in every possible way from a graph G, and then it can be asked how many (both minimum and maximum) of these subgraphs are required to reconstruct G up to isomorphism. Similar questions can also be posed for the less studied case of edge deletion. For graphs in certain classes there are known formulas to quickly determine reconstruction numbers. However, for the vast majority of graphs the computation devolves to brute force exhaustive search. Previous computer searches have found the 1-vertex-deletion reconstruction numbers of all graphs of up to 10 vertices. In this paper computed values of 1-vertex-deletion and 1-edge-deletion reconstruction numbers for all graphs on up to 11 vertices are reported. Several examples of graphs with high reconstruction numbers are also presented. This was made possible by an improved algorithm which enabled significant reduction in computation time.