Community detection in real-world networks is typically addressed through the use of graph clustering methods that partition the nodes of a network into disjoint subsets. While the definition of community may vary, it is generally accepted that elements of a community should be "well-connected". We evaluated clusters generated by the Leiden algorithm and the Iterative Kcore (IKC) clustering algorithm for their susceptibility to become disconnected by the deletion of a small number of edges. A striking observation is that for Leiden clustering of real-world networks, except for cases with large resolution parameter values, the majority of clusters do not meet the relatively mild condition we enforce for well-connected clusters. We also constructed a modular pipeline to enable well-connected output clusters that allows a user-specified criterion for a valid community considering cluster size and minimum edge cut size and describe the use of this pipeline on real world and synthetic networks. An interesting trend we observed is that the final clusterings on real-world networks had small node coverage, suggesting that not all nodes in a network belong in communities. * Minhyuk Park and Yasamin Tabatabaee contributed equally † Vidya Kamath Pailodi and Vikram Ramavarapu contributed equally ‡