Graph decomposition via edge edits into a union of regular graphs

Abstract

Suppose a finite, unweighted, combinatorial graph $G=(V,E)$ is the union of several (degree-)regular graphs which are then additionally 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 $(v,w)\in E$ has a higher degree $\mbox{deg}(w) > \mbox{deg}(v)$. 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 disjoint union of (distance-)regular graphs. The number of edge operations depends on the maximum degree and number of vertices with a higher degree neighbor but is independent of the size of $|V|$.