Algorithm Idea

Assume that we have already clustered a subset S of vertices. Let us denote their clustering by $\{C_1, ..., C_m\}$. We will extend the clustering $\{C_1, ..., C_m\}$ to include the elements of S', by putting each vertex $v\in S'$ into the cluster $C=argmax_{i}(\frac{\vert\{u \vert u\in C_i, (u,v) \in E\}\vert}{\vert C_i\vert})$.

After the extension $\{C_1, ..., C_m\}$ is the clustering of $S \cup S'$. Please note that during the extension procedure we do not add new clusters, thus the number m of clusters is unchanged.