Graph decomposition via edge edits into a union of regular graphs

Abstract

Suppose a graph $G=(V,E)$ is the union of several (disjoint) regular graphs which are then connected with a few additional edges. G will then have only a small number of vertices $v\in V$ with the property that one of their neighbors w has a higher degree. We prove the converse statement: if a graph has few vertices having a neighbor with higher degree and satisfies a mild regularity condition, then, via adding and removing a few edges, the graph can be turned into a union of disjoint regular graphs. The number of edge edits depends on the maximum degree and number of vertices with a higher degree neighbor but is independent of $\left|V\right|$.