The selection of nodes that can serve as cluster heads, local sinks and gateways is a critical challenge in distributed sensor and communication networks. This paper presents a novel framework for identifying a minimal set of nexus nodes to ensure full network coverage while minimizing cost. By formulating the problem as a convex relaxation of the NP-hard set cover problem, we integrate the graph theoretic centrality measures of node degree and betweenness centrality into a cost function optimized via a relaxed L1-norm minimization. The proposed approach is applicable to static and dynamic network scenarios and does not require location or distance estimation. Through simulations across various graph models and dynamic conditions, it is shown that the method achieves faster execution times (lower complexity) and competitive sparsity compared to classical greedy and genetic algorithms (GA), offering a robust, distributed, and cost-efficient node selection solution.