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.