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Wireless sensor networks (WSNs) achieving environmental sensing are fundamental communication layer technologies in the Internet of Things. Battery-powered sensor nodes may face many problems, such as battery drain and software problems. Therefore, the utilization of self-stabilization, which is one of the fault-tolerance techniques, brings the network back to its legitimate state when the topology is changed due to node leaves. In this technique, a scheduler decides on which nodes could execute their rules regarding spatial and temporal properties. A useful graph theoretical structure is the vertex cover that can be utilized in various WSN applications such as routing, clustering, replica placement and link monitoring. A capacitated vertex cover is the generalized version of the problem which restricts the number of edges covered by a vertex by applying a capacity constraint to limit the covered edge count. In this paper, we propose two self-stabilizing capacitated vertex cover algorithms for WSNs. To the best of our knowledge, these algorithms are the first attempts in this manner. The first algorithm is stabilized under an unfair distributed scheduler (that is, the scheduler which does not grant all enabled nodes to make their moves but guarantees the global progress of the system) at most O (n 2) step, where n is the count of nodes. The second algorithm assumes 2-hop (degree 2) knowledge about the network and runs under the unfair scheduler, which subsumes the synchronous and distributed fair scheduler and stabilizes itself after O (n) moves in O (n) step, which is acceptable for most WSN setups. We theoretically analyze the algorithms to provide proof of correctness and their step complexities. Moreover, we provide simulation setups by applying IRIS sensor node parameters and compare our algorithms with their counterparts. The gathered measurements from the simulations revealed that the proposed algorithms are faster than their competitors, use less energy and offer better vertex cover solutions. [ABSTRACT FROM AUTHOR]