## 3 System Model
### Network Model: Basic Definitions
The rotation of the GNs has a good effect on the network performance in terms of the selected performance metrics. In order to support our claim, we have conducted simulations with and without rotating GNs (refer to the simulation results section). Since this rotation has an effect on the energy consumption of the network, the GNs' rotation needs to be optimized on the expected energy cost of the bottle neck node. However, this requires much extra work. Thus, for now, we use a non-optimized/heuristic approach for the nodes rotation. Figure 1: AUV-aided architecture of a UWSN. One may argue that we do not need an acoustic model to know that a larger distance consumes more energy, because by minimizing the hop distance, we may end up transmitting over more hops. However, the overall proposed scenario is not only somehow different, but also more focused on efficient data gathering. The nodes do not directly communicate with the sink; rather, they follow a hierarchical architecture: MNs, GNs, AUV and sink. Here, our objectives are not limited to energy efficiency; rather, these include reliability, as well. Thus, we actually need the acoustic model, whose details are given in the previous paragraph. In the undirected graph \(G=(V,E)\), \(V\) is further divided into V\({}_{CDS}\) and V\({}_{non-CDS}\) sets, such that, V\({}_{CDS}\subset\)V and V\({}_{non-CDS}\subset\)V. We can define V of graph G as V = V\({}_{CDS}\cup\) V\({}_{non-CDS}\), where V\({}_{CDS}\) is a set of CDS nodes and V\({}_{non-CDS}\) is a set of non-CDS nodes; moreover, V\({}_{CDS}\)\(\cap\) V\({}_{non-CDS}\) = \(\emptyset\). Similarly, edges, E, are subdivided into E\({}_{CDS}\) and E\({}_{non-CDS}\), such that E = E\({}_{CDS}\)\(\cup\) E\({}_{non-CDS}\). E\({}_{CDS}\) is a set of edges between two CDS nodes. E\({}_{non-CDS}\) is a set of edges between non-CDS nodes or CDS and non-CDS nodes. Edges in E\({}_{CDS}\) are further divided into E\({}_{MST}\) and E\({}_{non-MST}\) (MST = minimum spanning tree). A dominating set (DS) of graph \(G\) is a subset \(D\) of \(V\), such that every vertex in \(V\backslash D\) is adjacent to at least one member vertex of D. A CDS is defined as a subset \(D\) of \(V\), such that any node in \(D\) can access any other node in \(D\) by a path that lies entirely within \(D\), such that \(D\) induces a connected sub-graph within \(G\). The CDS is developed by using the extended localized algorithm presented by [PERSON] and [PERSON] in [21]. After establishment of CDS, the MST of CDS is constructed. MST is constructed by using the Euclidean distance between nodes \(i\) and \(j\). After the MST, the Hamiltonian circuit (HC) is developed, which is a random trajectory of the AUV. The sub-optimal trajectory for the AUV leads to enhanced network throughput. The criterion for the eligibility of nodes to be included in CDS is shown in the flow chart. CDS formation includes the following steps:  
* First of all, check the degree of nodes. After that, consider those nodes that have the highest degree and then the second highest degree. The highest degree nodes are called dominator nodes. The nodes that are adjacent to the highest degree nodes are called dominatee nodes. * Dominatee nodes that are not further adjacent to any other node are called leaf nodes. * Dominatee nodes other than the leaf nodes are converted to dominators if that dominatee node has only leaf neighbours. * The node that is the dominatee of two dominator nodes is also converted to a dominator node. * The dominator nodes make the CDS in which every node is able to access any other CDS node by a path that exist entirely within the CDS, and every non-CDS node is adjacent to at least one CDS node. It seems logical that GNs would be (or close to) nodes in CDS (instead of rotating). However, if the GNs are not regularly rotated (on the basis of residual energy), then these will continuously consume relatively high energy that would lead to their earlier death. Afterwards, the far-away nodes will also consume high energy due to distant communication.