Missingvertex

Missingvertex is a term used in graph theory and data reconstruction to describe a vertex that is absent from a partial graph or dataset, and the associated problems and methods for inferring or reinstating that vertex. In theoretical contexts, it is often discussed alongside the graph reconstruction problem, which studies whether an original graph can be recovered from a collection of subgraphs obtained by deleting vertices, sometimes referred to as the deck. The missing vertex in this setting is the vertex whose removal yields a particular subgraph, and determining how to reattach or identify it is central to the reconstruction question.

A foundational idea linked to missingvertex is the Reconstruction Conjecture, proposed by Kelly and Ulam in 1952, which posits that graphs with at least three vertices are determined up to isomorphism by their vertex-deleted subgraphs. While the conjecture remains open in general, it motivates research into how much structural information—such as degree sequences, neighborhood patterns, and edge counts—can uniquely specify the missing vertex and its connections.

In practical terms, the concept appears in data and network applications where parts of a graph are incomplete or corrupted. Techniques drawn from missingvertex problems are used to infer missing links, verify data integrity, or guide reconstruction in sensor networks and graph databases. In software and algorithm libraries, missingvertex can also appear as a placeholder notion or identifier for an unknown vertex during graph-processing and validation tasks.

See also: vertex-deleted subgraph, graph reconstruction conjecture, graph isomorphism.