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Many degree sequences can only be realised in graphs that contain a`ds-completable card', defined as a vertex-deleted subgraph in which theerstwhile neighbours of the deleted vertex can be identified from theirdegrees, if one knows the degree sequence of the original graph. We obtainconditions on the degree sequence, such that graphs whose degree sequencesatisfies one of the conditions must contain such a card. The methods allow allsuch sequences on graphs of order up to 10 to be identified, and some fractionof the sequences for larger graphs. Among other applications, this can be usedto reduce the computational task of generating graphs of a given degreesequence without duplicates.

Graph reconstruction and generation from one card and the degree sequence