Degree-associated reconstruction number of graphs

A card of a graph G is a subgraph formed by deleting one vertex. The Reconstruction Conjecture states that each graph with at least three vertices is determined by its multiset of cards. A dacard specifies the degree of the deleted vertex along with the card. The degree-associated reconstruction number  drn ( G ) is the minimum number of dacards that determine G . We show that drn ( G ) = 2 for almost all graphs and determine when drn ( G ) = 1 . For k -regular n -vertex graphs, drn ( G ) ≤ min { k + 2 , n − k + 1 } . For vertex-transitive graphs (not complete or edgeless), we show that drn ( G ) ≥ 3 , give a sufficient condition for equality, and construct examples with large drn . Our most difficult result is that drn ( G ) = 2 for all caterpillars except stars and one 6-vertex example. We conjecture that drn ( G ) ≤ 2 for all but finitely many trees.