## 1 Introduction  
In the near future, the ocean will supply a substantial part of human and industrial needs: the oil and gas industry will move into deeper waters, the renewable energy will be harvested from sea, as well as many other innovative practices will become common. Furthermore, minerals such as cobalt,nickel, copper, rare earths, silver, and gold will be mined from the seafloor. One major challenge in this context is how to develop systems and solutions which can guarantee a sustainable development of the maritime activities so that this fragile habitat is protected for future generations. To this end, new offshore and port infrastructures need to be built, maintained, and repaired when necessary. Ocean monitoring and underwater exploration are not easy tasks, because the ocean is large and most of the underwater environment is still unknown to us. In addition, due to the high pressure in deep water, it is not suitable for people to work for long time under such conditions, which in several cases makes any human intervention impracticable. Thus, there is an obvious requirement for solutions that can assist humans in underwater mission. For this reasons, the Smart and Networking Underwater Robots in Cooperation Meshes (SWARMs) project (which can be found at [[http://swarms.eu/](http://swarms.eu/)]([http://swarms.eu/](http://swarms.eu/))) has been proposed to provide feasible solutions for these kinds of issues. The aim of the SWARMs project is to make autonomous underwater vehicles (AUVs), remote operated vehicles (ROVs) and unmanned surface vehicles (USVs) further accessible and useful, which includes: (1) enabling AUVs/ROVs to work in a cooperative mesh thus opening up new applications and ensuring reusability; (2) increasing the autonomy of AUVs/USVs and improving the usability of ROVs for the execution of simple and complex tasks. Therefore, for achieving the cooperation and communication between different AUVs, the AUVs must be able to exchange messages between each other, which signifies an efficient and reliable communication network is necessary for SWARMs. Since radio frequency (RF) waves are seriously attenuated in the underwater environment [1], underwater AUVs use acoustic waves rather than RF waves to communicate with each other [2]. In SWARMs, for providing an efficient and reliable communication network to facilitate mission execution, one of the important and necessary issues is the topology control of the underwater cooperating AUVs network. The reasons are: (1) to achieving their missions, underwater AUVs keep moving along the pre-defined paths, in which case, the topology of an underwater AUV cooperation network changes frequently; (2) since the transmission speed of acoustic waves is much smaller than that of RF waves, the transmission delay in an Underwater Cooperating AUVs Network (UCAN) is more critical than that in traditional RF-based wireless sensor networks (WSNs) [3; 4]; (3) due to the fact the bandwidth of the underwater acoustic channel is very limited, the congestion in such a kind of channel could be severe [5], which means the control messages of an underwater cooperating AUV network should be reduced as much as possible; (4) the multipath interference and the Doppler spreading are serious in the underwater environment. Although many topology control algorithms have been proposed for conventional WSNs, considering the control cost, these topology control algorithms are not efficient in an underwater cooperating AUVs network where the AUVs move frequently. For instance, in traditional topology control algorithms, once the current transmission power is not equal to the optimal one, which is calculated based on optimal algorithms (such as those described in [6; 7; 8; 9]), then the nodes need to adjust their transmission power. In a network where the network topology changes slightly or the network resources are abundant, the above mentioned approach is effective.