The degree-associated edge-reconstruction number of disconnected graphs   and trees
  An edge-card of a graph G is a subgraph formed by deleting an edge. The edge-reconstruction number of a graph G, ern(G), is the minimum number of edge-cards required to determine G up to isomorphism. A da-ecard is an edge-card which also specifies the degree of the deleted edge, that is, the number of edges adjacent to it. The degree-associated edge-reconstruction number, dern(G) is the minimum number of da- ecards that suffice to determine the graph G. In this paper we state some known results on the edge-reconstruction number of disconnected graphs and trees. Then we investigate how the degree-associated edge- reconstruction number of disconnected graphs and trees vary from their respective edge-reconstruction number. We show how we can select two da-ecards to identify caterpillars uniquely. Finally we conjecture that for any tree T, dern(T)<= 2. 
Kevin J. Asciak
math.CO