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Main components of internet of things such as wireless sensor networks (WSNs) are generally modeled as graphs. Matching, dominating set, spanning tree, and vertex cover are fundamental graph theoretical structures which are widely used to solve backup assignment, clustering, backbone formation, routing, link monitoring, and other important problems in WSNs. Capacitated versions of these graph theoretical problems restrict at least one feature of the original problem such as a capacity value assignment to restrict the number of cluster member that a cluster head can serve in capacitated dominating set problem, limiting the number of nodes in each subtree connected to the root in capacitated minimum spanning tree problem, etc. The general aim of these operations is to provide energy efficiency by balancing the load evenly across the nodes. In this chapter, the authors introduce important capacitated graph theoretical problems and explain both central (sequential) and distributed algorithms for constructing these structures. The operations of the algorithms are demonstrated by sample topologies.