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The method that involves each router sharing its routing table with neighbors is Distance Vector Routing. In this routing approach, routers periodically send their entire routing table to their directly connected neighbors. This means that each router communicates not just information about new routes or changes but shares the full list of all possible routes and their associated distances (or costs) to every other router in its routing table. This sharing of information allows routers to update their own routing tables based on the information received from neighbors.

Distance Vector Routing operates on the principle of "tell your neighbors," meaning that each router relies on the information it receives from its peers to update its own view of the network. This continues until routing information converges and all routers have a consistent view of the network topology.

In contrast, Link State Routing uses a different methodology, where routers instead share information about the state of their own links and stipulate the status of their connections rather than sharing entire routing tables. Hybrid Routing combines features of both Link State and Distance Vector protocols, and Exterior Gateway Protocol primarily pertains to routing between different autonomous systems, focusing on route exchange on a broader scale. Thus, the characteristic of sharing entire routing tables distinctly identifies Distance Vector Routing.