## 3 Methodology
### Traffic Conflict Graph and Communication Topology  
The complexity of the conflict types at intersections determines the difficulty level of traffic control. Typical intersections have 16 conflict points, as shown in Figure 1, which implies that vehicles from different approaching lanes may crash at these points. In addition, each vehicle is under a conflict relationship with its immediately preceding or following vehicle on the same lane, i.e., rear-end collision. We construct a conflict graph to describe the conflict relationships of the approaching lanes, as shown in Figure 2. Define the set of approaching lanes as \(\mathcal{L}\), where \(\mathcal{L}=\{L_{1},L_{2},L_{3},L_{4},L_{5},L_{6},L_{7},L_{8}\}\), and the conflict set \(\mathcal{C}_{h},h\in\mathcal{L}\) denotes the set of lanes that has conflict relationships with lane \(h\). Take the westbound straight lane \(L_{1}\) as an example, \(\mathcal{C}_{L_{1}}=\{L_{1},L_{3},L_{6},L_{7},L_{8}\}\), which means that the trajectories of vehicles on lane \(L_{1}\) may conflict with those on lanes \(L_{1}\), \(L_{3}\), \(L_{6}\), \(L_{7}\), and \(L_{8}\) at the intersection. Conversely, vehicles on lane \(L_{1}\) have no risk of collision with other vehicles on the lanes outside the set \(\mathcal{C}_{L_{1}}\).  
For eliminating collisions at intersections, a reasonable idea is to allow vehicles on the conflicting lanes to cross the intersection asynchronously while keeping a safe distance between each other. From the point of view of a single vehicle, no collision will occur if it retains a virtual safe car-following spacing with all potentially conflicting vehicles. We call these vehicles on the lanes in the conflict set the neighboring vehicles of the ego vehicle. The ego vehicle must exchange information with its neighbor vehicles in real time through V2V communication to maintain a virtual safe car-following spacing. Considering all vehicles in the CZ, the communication topology of vehicles can be modeled by a time-varying graph \(\mathcal{G}_{t}=\{\mathcal{S}_{t},\mathcal{E}_{t}\}\), where \(\mathcal{S}_{t}=\{1,2,\ldots,n_{t}\}\) is the set of nodes (vehicles), and \(\mathcal{E}_{t}\subseteq\mathcal{S}_{t}\times\mathcal{S}_{t}\) is the set of edges (information flowing directions). The characteristic of graph \(\mathcal{G}_{t}\) can be modeled by an adjacent matrix, which is defined as \(\mathcal{A}_{t}=[a_{ij}(t)]\in\mathcal{R}^{n_{t}\times n_{t}}\) with  
\[\begin{cases}a_{ij}(t)=1,\text{if }\{j,i\}\in\mathcal{E}_{t}\\ a_{ij}(t)=0,\text{if }\{j,i\}\
otin\mathcal{E}_{t}\end{cases}\quad i,j\in \mathcal{S}_{t} \tag{2}\]  
where \(\mathcal{R}^{n_{t}\times n_{t}}\) represents the set of \(n_{t}\times n_{t}\) real matrices, \(\{j,i\}\in\mathcal{E}_{t}\) denotes that vehicle \(i\) can receive the information of vehicle \(j\). Define the neighboring set of node \(i\) as \(\mathcal{N}_{i}(t)\), where \(\mathcal{N}_{i}(t)=\{j|a_{ij}(t)=1,j\in\mathcal{S}_{t}\}\). In other words, node \(j\) is called a neighbor of node \(i\) if and only if \(a_{ij}(t)=1,j\in\mathcal{S}_{t}\). Correspondingly, we define a dual set \(\mathcal{D}_{i}(t)=\{j|a_{ji}(t)=1,j\in\mathcal{S}_{t}\}\), which represents that any node \(j\in\mathcal{S}_{t}\) can receive the information of node \(i\). Obviously, we have \(\mathcal{N}_{i}(t)=\mathcal{D}_{i}(\mathbf{t})\) under an undirected graph.  
Figure 2: Traffic conflict graph for the intersection in Figure 1.