## 3 Network Models and Problem Description
### Problem Description  
In the process of solving the problem of this scheme, the backup paths of the nodes are subjected to constant search, and therefore, the connectivity of the graph needs to be ensured, i.e., it has the following definition.  
**Definition 3** (Connectivity of a graph).: _If a topological graph \(G=(V,E,W)\) is connected, i.e., for any two nodes \(s,d\in V\), there exists a path whose sequence of nodes connects \(s\) and \(d\), it can be expressed by Equation (3), as follows:_  
\[\partial(G)=\{\forall s,d\in V\exists v_{1},v_{2}...v_{k}|v_{1}=s,v_{k}=d\} \tag{3}\]  
_Furthermore, for any \(1\leq i\leq k\), there is \((v_{i},v_{i+1})\in E\), which satisfies the above condition, and then the topological graph is connected [14]. Connectivity is the basis for determining whether a path exists from the start point to the end point._  
The main aim of this paper is to solve a central problem in intra-domain route protection as follows: in a given undirected graph topology, using the shortest path algorithm, determine one backup path for each pair of source node \(s\) and destination node \(d\). This backup path should satisfy the minimum cross-degree with the best path, where \(sp(s,d)\) is denoted as the best path from node \(s\) to node \(d\), \(sp^{\prime}(s,d)\) is denoted as the backup path, and \(sp(s,d)\cap sp^{\prime}(s,d)\) denotes that the cross-degree of the two paths is denoted by the symbol \(C\). At this point, the problem can be simplified and expressed as \(\min K(C)\). The minimum cross-degree is beneficial for reducing the risk of a single point of failure in the network, while improving the stability of the network in the face of failure. Therefore, the scheme needs to fulfill the following conditions:  
1. The optimal path has minimum cross-degree with the backup path, or the backup path and the optimal path are two independent paths.
2. According to the shortest path algorithm ([PERSON]'s algorithm), the source node forwards the packet to the destination node via the best path or backup path.
3. The connectivity of the topology needs to be maintained during the computation of backup paths.  
Figure 1: Topological map.  
The solution to the problem proposed in this paper must consistently satisfy the aforementioned three conditions, while simultaneously minimizing the intersection between the backup path and the optimal path during the backup path search process. Therefore, the problem can be formulated as a Linear Programming (LP) problem, namely,  
\[\min K(C)=\min\{sp(s,d)\cap sp^{\prime}(s,d)\} \tag{4}\]  
s.t.  
\[\forall G=(V,E,M)\in UG \tag{5}\]  
\[G\rightarrow\{\partial(G)|\forall 1\leq i\leq k,(v_{i},v_{i+1})\in E\} \tag{6}\]  
\[sp(s,d)\&sp^{\prime}(s,d)=Dijkstra(G,s,d) \tag{7}\]  
\[sp(s,d)=s,s_{1},s_{2}...d \tag{8}\]  
\[K(C)=\{sp(s,d)\cap sp^{\prime}(s,d)\} \tag{9}\]  
\[sp^{\prime}(s,d)=\{(d_{i},d_{i+1})\in E/C\} \tag{10}\]  
In the aforementioned LP model, the objective function is defined by Equation (4), where the cross-degree must be minimized under the constraints imposed by Equations (5)-(10). The term \(UG\) in Equation (5) denotes an undirected graph, confirming that the initial topology of this scheme adopts an undirected graph structure. Equation (6) uses \(\partial(G)\) to represent graph connectivity, fulfilling one of the essential requirements--preserving the connectivity of the topology graph. Both the optimal path and the backup path in this study are derived using [PERSON]'s shortest path algorithm (Equation (7)). Equation (8) specifies the optimal path sequence between \(s\) node and \(d\) node. The intersection sequence of the optimal path \(sp(s,d)\) and the backup path \(sp^{\prime}(s,d)\) is characterized by Equation (9), which quantifies their overlapping path segments. Equation (10) imposes constraints on the backup path, ensuring that the backup path from node \(s\) to node \(d\) remains mutually independent from its corresponding optimal path.