## 1 Introduction  
Throughout industry and academia, wireless sensor networks (WSNs) have gained much attention in recent years. Comprising small, low power, limited-capacity devices, with data sent to a sink node on a many-to-one multi-hop basis. Given the environments in which these networks may be located, frequently inhospitable, with applications in many areas such as deep-sea oil and gas [1], disaster recovery [2], and agriculture [3; 4], it may not always be possible to replace the batteries in network devices [5]. As a result, power conservation techniques which maximise node and, subsequently, network lifetime, take on importance. A common solution within WSNs to this issue is to duty cycle node power, such that when idle the nodes shall be asleep. However, whilst of benefit in terms of power consumption, duty cycling gives rise to other issues within WSNs. Heterogeneity of wake-up schedules within nodes in a WSN, as a result of duty cycling, results in problems with regard to neighbour discovery (ND), such that it becomes a challenge to ensure these schedules overlap in order that nodes may discover each other and data may be transmitted between them. This issue can then be exacerbated if mobility is also a factor within the network, ND schemes also now having to allow for nodes being within the same vicinity as well as an overlap of wake-up schedule. This may occur when the sink node is mobilised in order to combat the particular issue of energy and routing hotspots [6] in static WSNs. These occur near the sink node as nodes closest to the sink will assume a greater responsibility in terms of routing than leaf nodes and, as a result, consume more energy. Ultimately, the lifetime of these hotspot nodes can be reduced considerably, with the other nodes in the network left unable to communicate with the sink node. Therefore, the WSN itself ceases to function properly. In utilising a mobile sink node (MSN), moving around or across the network, energy consumption is spread more evenly amongst the nodes in the WSN. This is due to no particular node being able to take on the role of hotspot for a considerable period of time. As a result, network lifetime can be increased and the hotspot issue negated to a certain degree [7; 8]. The mobilising of sink nodes is possible via many different applications such as robots, vehicles, or being located on a person. However, the emergence of unmanned aerial vehicles (UAVs), commonly referred to as drones, provides further possibilities in this area, with studies having taken place to utilise drones in this way, to collect data and provide network connectivity [9]. As previously stated though, mobility adds another layer of complexity when considering ND and duty cycling. Sink mobility is no exception to this. Whereas in the use of MSNs, studies have taken place to extend existing work or, more specifically, develop new network layer routing protocols, thus far none have taken account of ND protocols. However, the issue of ND with regard to the mobility of devices in general in the internet of things (IoT), and specifically WSNs, has seen a great amount of research. With ND protocols in WSNs generally more concerned with duty cycling at the media access control (MAC) layer, referencing ND as a network layer issue has been designed to older, more traditional fixed networks. In networks consisting of small, battery-powered and duty cycled [10] devices, ND is no longer used to refer to IP addressing and duplicate address detection [11]. Now the greater concern is ensuring that ND is possible at all, with the overlap of wake-up schedules essential in this aim. In order to achieve this in an energy-efficient way, many probabilistic approaches have been proposed [12; 13]. These techniques have been shown to be efficient, however, there is also the possibility that probabilistic methods may result in the log-tail discovery issue. This is where it may be the case that a node is not discovered at all [14]. As such, deterministic [15; 16; 17] algorithms occur more frequently in research.