## 1 Introduction  
With the rapid development of network devices and the surge in network traffic, these devices now combine multiple functions and can provide various services simultaneously to mobile users. This results in multiple concurrent data flows. When a mobile device moves, the data transmission path in the wired network may need to be updated due to a change in the device's access switch. It is important to note that, in this paper, the term \"wired network\" refers to a static network composed of switches and links. Traditional mobility management schemes, whether anchor-based or non-anchor-based, typically select a single switching path. However, when this path cannot accommodate all the mobile data flows, it may lead to network congestion, resulting in increased packet delays and higher packet loss rates. Such issues severely degrade the user experience, especially for real-time services like Voice over Internet Protocol (VoIP) and online gaming, which require low transmission delays, and for video live streaming, where high packet loss rates can cause audio-video synchronization issues. Recent research has explored providing multiple switching paths for mobile data flows by selecting different late-binding nodes for each flow [1]. However, this approach does not take into account the potential overlap of links between paths, which limits its effectiveness in complex networks. To address this limitation, multi-topology routing can be employed. This approach generates multiple logical topologies as subsets of the physical network topology. When the mobility management process is triggered and a new network address of the mobile device is obtained, a logical topology is selected for each mobile data flow. This method enables the distribution of flows across multiple paths, reducing the possibility of congestion caused by relying on a single switching path. In this scheme, the construction of logical topologies plays a critical role, as it provides the foundation for the subsequent topology selection process. Well-designed topologies can provide high-bandwidth, low-latency paths to all possible destination nodes, significantly improving the final performance of mobile users. Conversely, poorly designed topologies fail to provide such paths and waste the limited flow entry capacity of switches. Previous studies on multi-topology construction have mainly focused on other objectives. Some are motivated by node or link failure recovery [2; 3; 4; 5]. These studies aim to quickly restore transmission by switching topologies when failures occur. Others focus on traffic engineering [6; 7; 8; 9], aiming to maximize link utilization by increasing path diversity or employing other strategies. Other studies aim to isolate different types of flows and allocate them to separate topologies to provide varying levels of Quality of Service (QoS) [10; 11; 12]. However, these methods are not suitable for providing multiple switching paths specifically for mobile data flows. Mobile devices may move in and out of the coverage area of access points connected to various access switches. Therefore, the multi-topology construction must consider multiple paths from the source node (e.g., the switch connected to a content server) to each access switch. Moreover, the handover mechanism in wireless networks is outside the scope of this study. We focus solely on optimizing switching paths in wired networks. The number of logical topologies deployed in the physical network must also be limited, as it impacts both switch capacity utilization and the computational complexity of selecting topologies for data flows, ultimately affecting the performance of mobile users. For a given source-destination node pair, multiple paths may exist within different logical topologies. If adding a new path does not increase the total available bandwidth between the node pair, it is considered redundant. Including such paths unnecessarily increases the number of logical topologies and wastes switch capacity.