Degree Lists and Connectedness are 3-Reconstructible for Graphs with At Least Seven Vertices

The k-deck of a graph is the multiset of its subgraphs induced by k vertices. A graph or graph property is l-reconstructible if it is determined by the deck of subgraphs obtained by deleting l vertices. We show that the degree list of an n-vertex graph is 3-reconstructible when n≥ 7 , and the threshold on n is sharp. Using this result, we show that when n≥ 7 the (n- 3) -deck also determines whether an n-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are 2-reconstructible when n≥ 6 , which are also sharp.