## 4 Topology Control
### Post-Deployment
#### 4.3.1 Connectivity, Energy-Efficiency and Coverage  
In addition to connectivity, there are several other issues that should be taken into account, such as the lifetime of nodes. Higher transmission power results in faster energy consumption. For example, network connectivity is the largest when each node transmits at its maximum power, but it has the shortest lifetime. Extending the lifetime of the network in some scenarios is a desirable requirement, since the sensor nodes may be deployed in areas where humans are not allowed to enter, or the nodes cannot be replaced, or the battery cannot be recharged. Besides, if there are many redundant nodes, the coverage problem arises when scheduling some nodes to be active and others to be inactive, which is a common approach to maintain the network connectivity and coverage. Figure 7: Construct a new \(2-connected\) network \(H\) from the existing \(2-connected\) network \(G\). (1) Adjust Transmission Range:  
The purpose is to assign each node a transmission power so that the resultant network is \(k-connected\), and meanwhile, the transmission power is optimized. Optimizing energy while keeping network connectivity strong is a challenging task [8, 34, 81, 82, 83]. Most of the energy optimization and \(k-connected\) problems have been proven to be NP-complete [34]. Note that minimizing the total transmission power assignment is NP-hard even for a \(1-connected\) network [84, 85]. There are centralized, distributed, or hybrid (e.g., [84]) approaches to address this problem. If the node transmits with maximum transmission power and finds out the \(k\) optimal vertex disjoint path to each neighbor according to certain criteria (e.g., power cost), [86] proves that if each node maintains \(k\) optimal disjoint paths, the resultant network is \(k-connected\) globally, provided that all nodes use the maximum transmission power. In [84], the \(k-connected\) network is formed from three steps: first, the network becomes clustered; second, the cluster head assigns each node a communication power, such that each node in the cluster is \(k-connected\) (this is called the intra-cluster topology control); third, it is ensured that there are \(k-disjoint\) paths between adjacent clusters (this is called inter-cluster topology control). After these three steps, the resultant network is \(k-connected\) globally. The work in [87] aims at building up and maintaining \(k-regular\) and \(k-connected\) nodes by adding links between the overlay node. This approach forms groups consisting of \(k\) nodes, and each group is a \(k-regular\) network; it regards the group as a single node, then recursively using the same approach to form the final topology. If the network already is a \(k-connected\) network, what is the condition preserving \(k-connected\) by changing the power based on the angle of neighbors? The work in [9] studies the angle of its neighbors. The cone of the degrees of nodes determines the connectivity. A node \(u\) transmits with the minimum power \(P\), in order to ensure that in every cone of the degrees around \(u\), there are some nodes that \(u\) can reach with power \(P\); it is shown that taking angle \(\frac{5\pi}{6}\) is a necessary and sufficient condition to guarantee that the network connectivity is preserved. The work in [9] tries to use fewer edges by removing edges, but preserves connectivity. Two basic mechanisms are adopted [9]: (1) start from the initial power, then increase the power until connected; and (2) start from the largest energy, then reduce it as long as the connectivity is not changed. The work in [81] preserves the connectivity of the network and meanwhile minimizes the transmission power. The work in [81] proves that the upper bound of the angle is \(\alpha\leq\frac{2\pi}{3k}\). Note that many algorithms try to optimize the energy, inevitability incurring extra overhead. (2) Schedule the Duty Cycle of Nodes:  
Another common method to optimize energy is to schedule the duty cycle of nodes.