Graph reconstruction and generation from one card and the degree sequence

TL;DR

This paper explores conditions under which a degree sequence guarantees the existence of a special vertex-deleted subgraph, aiding in graph reconstruction and generation from degree sequences.

Contribution

It introduces new conditions on degree sequences that ensure the presence of a ds-completable card, improving graph reconstruction methods.

Findings

Identifies conditions on degree sequences for ds-completable cards.

Classifies all such sequences for graphs up to 10 vertices.

Provides partial classification for larger graphs.

Abstract

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