Link-Cut trees accelerate max flow algorithms: In Dinic's algorithm: Find blocking flow faster by maintaining the level graph as a Link-Cut tree. In Push-Relabel: Track excess flow routing through dynamic tree structure.
Sleator-Tarjan dynamic trees were originally invented for max flow, achieving for flow in networks with edges. Here's the trick: augmenting path operations become instead of when using Link-Cut trees for the residual graph structure.