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 $|V|$.

翻译：假设一个有限、无权的组合图 $G = (V,E)$ 是若干个（度）正则图的并集，并且这些正则图通过少量额外边相互连接。那么 $G$ 中只有少数顶点 $v \in V$ 满足：其某个邻点 $(v,w) \in E$ 的度数 $\mbox{deg}(w) > \mbox{deg}(v)$。我们证明其逆命题：若一个图中存在少量顶点具有度数更高的邻点，且满足一个温和的正则性条件，则通过添加和删除少量边，该图可转化为若干（距离）正则图的不交并集。边操作的数量取决于最大度数及具有更高度数邻点的顶点数，但与 $|V|$ 的大小无关。