Modifying a Graph's Degree Sequence and the Testablity of Degree Sequence Properties

TL;DR

This paper demonstrates that graphs with degree sequences close to a realizable sequence are also close in structure, enabling the testing of degree sequence properties efficiently in dense graphs.

Contribution

It establishes a connection between degree sequence proximity and graph edit distance, and proves the testability of degree sequence properties in dense graph models.

Findings

Graphs close in degree sequence are close in edit distance.

Degree sequence properties are testable with query complexity independent of graph size.

Provides a new method for property testing based on degree sequence proximity.

Abstract

We show that if the degree sequence of a graph $G$ is close in $\ell_1$-distance to a given realizable degree sequence $(d_1,\dots,d_n)$, then $G$ is close in edit distance to a graph with degree sequence $(d_1,\dots,d_n)$. We then use this result to prove that every graph property defined in terms of the degree sequence is testable in the dense graph model with query complexity independent of $n$.