# Topology Control of Low-Connection UAV Laser Network with Virtual Nodes
## 3 Model  
The UAV laser network discussed in this paper consists of \(n\) nodes distributed in a two-dimensional plane. Each node is equipped with four laser communication modules, all of which have equal maximum communication ranges. Each module can establish a communication link with another module, and the coverage area of a single laser communication module is a circle with a radius equal to its maximum communication distance. Additionally, each node can determine its own location through GPS or other means. Since the actual size of the nodes is much smaller than the maximum communication distance of the laser communication modules, the maximum communication distance can be considered as the communication range of the nodes. For convenience in description, we normalize the maximum communication distance of the nodes to 1. Thus, we can model this network using the commonly used unit disk graph (UDG) in ad hoc networks. For the \(UDG=(V,E)\), let  
\[V=\left\{v_{i}|v_{i}\in R^{2},i=1,2\cdots n\right\}\]  
represent the set of nodes and  
\[E=\left\{e_{k}|\forall u,v\in V,u\
eq v,\ \|uv\|_{2}\leqslant 1\right\}\]  
represent the set of edges. Clearly, since each UAV is equipped with only four laser communication modules, the degree of each node is limited to a maximum of 4. The XTC algorithm proposed in Ref. [18] can reduce the upper bound of the node degree to 6, whereas the required upper bound for the node degree is 4. Considering connecting two nodes and treating them as a single node, each node would then have a remaining degree of 3, making the degree upper bound of this new node exactly 6. Therefore, based on the aforementioned approach, one can utilize the XTC algorithm to construct a subgraph \(G^{\prime}(V)\) with a degree upper bound of 4. Assume that the communication radius of the new node (hereafter referred to as the virtual node) is \(r^{*}\) and graph \(G\) remains connected under the condition that the communication radius \(r<1\). Let the new graph formed by virtual nodes be denoted as the virtual graph \(G^{*}(V^{*},E^{*})\), and let the subgraph of graph \(G\) at communication radius \(r\) be denoted as \(G^{\prime}_{r}\). Graphs \(G\), \(G^{\prime}_{r}\), and virtual graph \(G^{*}\) need to satisfy the following conditions:  
1. If subgraph \(G^{\prime}_{r}\) is connected, then the new graph obtained after pairing nodes should also be connected. That is, for the virtual graph \(G^{*}\), at a communication radius of \(r^{*}\), \(G^{*}\) should be a connected graph. This implies that the process of creating virtual nodes by pairing existing nodes in \(G^{\prime}_{r}\) does not disrupt the overall connectivity of the graph, and the virtual graph \(G^{*}\) inherits the connectedness property from \(G^{\prime}_{r}\). 2. For every edge in the virtual graph \(G^{*}\), there must be at least one corresponding edge in the original graph \(G\). This ensures that the virtual connections in \(G^{*}\) are grounded in actual connections within \(G\), maintaining a reflection of the original graph's structure and connectivity within the virtual framework. Assuming that the virtual node is located at the midpoint of the line segment connecting the paired nodes, the maximum distance from the virtual node to either of the two nodes would be \(r/2\). To ensure the connectivity of the virtual graph, it is assumed that the communication range of the virtual node is contained within the common communication range of the paired nodes. Additionally, the paired nodes corresponding to neighboring virtual nodes should also be located within the common communication range, as illustrated in Figure 2. Under these conditions, Condition 2 is necessarily satisfied.