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 : v ∈ V ( G ) }. Myrvold (1992) showed that the degree sequence of a graph on n ≥ 7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can be 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.