## 1 Introduction  
Oceans cover about two-thirds of the surface of the earth. They have gained importance because of the hidden resources, military surveillance purposes and underwater environment monitoring. UWSNs have gained attention for these explorations. Typical framework of UWSN is composed of underwater sensor nodes, sink nodes, surface stations, _etc._ Acoustic signals are used for communication in UWSNs due to the relatively low absorption rate in underwater environment. However, continuously varying water temperature, salinity, pressure, density and noise make the underwater environment harsh for signal propagation. Furthermore, the unique characteristics of acoustic signals impose numerous challenges on devising efficient data routing protocols in UWSNs. Sensor nodes in UWSNs consume more energy due to severe environmental conditions where replacement or recharging of battery is not feasible. Bandwidth available for communication is limited (<100 kHz) because of high attenuation and absorption of acoustic signals. Link characteristics like transmission loss, propagation delay and Signal-to-Noise Ratio (SNR) are continuously varying with time and depth. Moreover, BER is also high due to multi-path fading and refraction phenomena. Therefore, improved network lifetime, energy efficiency and high throughput along with reliability of received data stand out as a critical consideration [1]. Graph theory serves to model different network topologies. It is widely used to devise algorithms for efficient data routing in Wireless Sensor Networks (WSNs). UWSNs involving depth-based routing can be represented by a directed graph (or digraph), \(G=(V,A)\). A digraph is a graph, or set of nodes \((V)\) connected by directed edges or arcs \((A)\), where the arcs have a direction associated with them. We are considering a digraph for data routing in underwater environment because in depth-based routing data needs to be routed upwards, that is, towards low depth (surface station present on water surface). Cooperative communication exploits the broadcast nature of wireless transmission and relies on cooperation of nodes in relaying each others' information. It involves the use of a source, relay and destination node. When source node broadcasts any signal it is received by destination node and also overheard by relay node in its sensing range. Relay node then forwards this overheard signal to destination node as a replica of the original signal [1]. Two types of relaying schemes usually implemented at relay nodes are: Amplify-and-Forward (AF) and Decode-and-Forward (DF) [2]. Relay node simply amplifies the signal and transmits it in the former scheme, whereas, received signal is decoded, corrected, recoded and then transmitted in later. Different number of relay nodes can be selected as per application requirement. Cooperative communication is incorporated in UWSNs to improve the network performance in terms of packet delivery ratio and data reliability. General architecture [3; 4; 5; 6] of UWSNs involves static sinks which are usually deployed on water surface. However, static sinks result in hotspot problem in which sensor nodes closer to the sinks tend to die early because of forwarding more packets (packets from nodes that lie far away from sinks), leaving certain area of network completely unmonitored. Hence, it is beneficial to use Mobile Sinks (MSs) to transfer sensed information [7]. In this paper, we present a mathematical model for cooperative routing along with sink mobility in UWSNs. This work is an extension of the work in [8] in which cooperative routing along with sink mobility is investigated at network layer. The proposed mathematical model involves an algorithm to compute the Dominating Set (\(DS\)) of a given network modeled as a digraph. \(DS\) formation helps in organizing nodes in a way that is optimal for cooperative routing. \(DS\) is a subset of vertices \(V\) such that each vertex in \(V-DS\) is adjacent to at least one vertex in \(DS\).