The low degree heuristic

When the input graph contains low degree vertices, one iteration of a minimum cut algorithm may simply separate a low degree vertex from the rest of the graph. This is computationally very expensive, not informative in terms of clustering, and may happen many times if the graph is large and sparse. Removing low degree vertices from the original graph before running the HCS algorithm eliminates such iterations and significantly reduces the running time. The complete algorithm, after refinements, is shown in figure 11.8.

  

Figure: Refinements of the HCS algorithm. $d_1,d_2,\ldots,d_p$ is a decreasing sequence of integers given as external input to the algorithm.

  

Figure 11.9: Results of HCS+Cluster Merging: T: True clusters; C: HCS+merge. (i,j): number of common elements in C(i) and T(j). 13 out of 16 clusters are almost pure, 6 are completely pure.