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Top1. Introduction
Wireless sensor networks (WSNs) are usually made up of hundreds even thousands of distributed sensor nodes by self-organized and multi-hop manner. WSNs can be used to perceive, collect, process and transmit information of the sensed objects, then send them to the owner of the networks (Elson & Estrin, 2004; Huang & Chu & Zhang, 2017). The sensor nodes are mostly battery operated and they are always deployed in harsh environments, so the battery replacement is impractical, and the sensor nodes are easy to encounter hardware faults or deliberate attacks (Emeka & Abraham, 2011; Aziz, Sekercioglu, Fitzpatrick, & Lvanovich, 2013). Hence, energy and fault-tolerant problems are two important concerns in the research of WSNs.
Topology control is an important method to solve energy and fault-tolerant problems and many energy-efficient and fault-tolerant topology control algorithms for WSNs have been presented. In Zhu, Luo, Peng, Li, and Qun (2009), a topology control algorithm was proposed to construct an energy-efficient and fault-tolerant topology for multi-hop communications in a two-tier WSNs. By constructing a k-regular overlay graph, the algorithm can ensure the connectivity of the network even with a small number of failure nodes. In Ishizuka and Aida (2004), and Li, Chu, and Wu (2017), a deployment algorithm was proposed to use limited sensor nodes as relay nodes to construct an energy-efficient and fault-tolerant topology. The topology constructed above could not only keep the connectivity between the sensor nodes and the sink node but also improve the survivability of WSNs. In Thallner and Moser (2005), and Yan, Song, Yang, and Yang (2015), a fault-tolerant topology control algorithm for k-disjoint paths was proposed. It could improve network fault tolerance by constructing k-disjoint paths from the sensor nodes to the sink node and adding redundant links. Although most of the above solutions can improve network tolerance, the existence of redundant nodes or k-connected networks degrades the network performance and affects the network lifetime.