## 2 Related Works  
Connectivity is a very important issue in underwater acoustic directional sensor networks (UADSNs); some related enlightening works are introduced in the following section.  
In [19], [PERSON] provided a detailed analysis of a cone-based distributed topology-control algorithm, which only depends on the directional information that a node can achieve. The basic idea is to transmit with the minimum power required to ensure that in every cone of degree \(\alpha\) around a node \(u\), there are some nodes that \(u\) can reach. It is shown that taking \(\alpha=5\pi/6\) is a necessary and sufficient condition to guarantee that the network is connected.  
In [20], [PERSON] et al. researched, under the condition of an unbounded transmission range, the smallest angle \(\alpha\) for which there exists an integer \(n=n(\alpha)\) such that for any node set \(\mathcal{P}\) of \(n\) antennas of angle \(\alpha\), there can form a symmetric communication graph by appropriately orienting the antennas. Finally, they came to the conclusion of \(n=4\) for \(\alpha=\pi/2\), and a thesis was further derived, which reveals that if an omni-directional unit disk connected graph meets the condition of \(r=14\sqrt{2}\), the induced symmetric communication graph connects.  
A new graph structure called \(\alpha\)-MST was studied in [21], which is a minimum spanning tree of a set of vertexes \(\mathcal{P}\) with the additional property that for each point \(p\in\mathcal{P}\), the smallest angle around \(p\) containing all the edges adjacent to \(p\) is at most \(\alpha\). A generation algorithm is provided in the paper for \(\alpha\)-MST, where a boundary of \(\alpha=\pi/3\) is asserted, while an \(\alpha\)-MST does not always exists when \(\alpha<\pi/3\). In addition, the author reduced the connectivity with \(\alpha=2\pi/3\) to be NP-hard from HPHGG, whose thinking afforded us a lesson in the proof of MNCTO in this paper.  
Ref. [22] studied an interesting relevant geometric problem from the view of mathematics; the authors formulated a problem of building a connected communication network for a set \(\mathcal{P}\) of \(n\) vertexes with an \(\alpha\)-degree directional antenna, and proved that it is always possible if \(\alpha\geq 60\). An \(O(n\log k)\) complexity algorithm was also proposed to orient the antenna direction, which utilized the properties of the convex hull of \(\mathcal{P}\). Nevertheless, given an unbounded transmission range, the network sets up without considering the energy consumption and leads to a geometric topology with a large average link length.  
[PERSON] et al. in [23] conducted some research on the work mentioned above, and showed that for antenna beamwidth \(\alpha<\pi/3\), the problem is NP-complete to connect the network with a given radius, while for \(\pi/3\leq\alpha<\pi/2\), it is still unknown how to connect any set of vertexes with a constant range.  
Some other related works helpful to our work are summarized as follows Table 1.  
Motivated by the observations, including, but not limited to, the results tabulated above, whose ranges seldom cover the condition of \(\alpha=\pi/3\), which coincides with our previous study on the most suitable beamwidth for an underwater directional transducer, we attempt to conduct some further work on the coincidence brought by the specialty of an underwater acoustic environment, which is illustrated in the next section.