Application: Max Flow

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 $O(mn \log n)$ for flow in networks with $m$ edges. Here's the trick: augmenting path operations become $O(\log n)$ instead of $O(n)$ when using Link-Cut trees for the residual graph structure.