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Clustering is a fundamental task in unsupervised learning, one that targets to group a dataset into clusters of similar objects. There has been recent interest in embedding normative considerations around fairness within clustering formulations. In this paper, we propose 'local connectivity' as a crucial factor in assessing membership desert in centroid clustering. We use local connectivity to refer to the support offered by the local neighborhood of an object towards supporting its membership to the cluster in question. We motivate the need to consider local connectivity of objects in cluster assignment, and provide ways to quantify local connectivity in a given clustering. We then exploit concepts from density-based clustering and devise LOFKM, a clustering method that seeks to deepen local connectivity in clustering outputs, while staying within the framework of centroid clustering. Through an empirical evaluation over real-world datasets, we illustrate that LOFKM achieves notable improvements in local connectivity at reasonable costs to clustering quality, illustrating the effectiveness of the method.