Reconstructing the degree sequence of a sparse graph from a partial deck

The deck of a graph $G$ is the multiset of cards ${G-v:vin V(G)}$. Myrvold (1992) showed that the degree sequence of a graph on $ngeq7$ vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree $d$ can reconstructed from any deck missing $O(n/d^3)$ cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.