The branching on fractional variables to achieve integrality is what guides the Branch-and-Price-and-Cut process, finally making sure that the solution is not just a valid solution to the LP relaxation, but also to the original MIP formulation.

During the execution of Branch-and-Price-and-Cut, GCG will select nodes to open and select variables to branch on. The whole process can be lead by tree heuristics (diving heuristics) that heuristically determine a good path to an integer solution. Most importantly, there are some things one has to pay attention to (GCG uses two copies of the tree), which will be explained in the page on the branching process. Then, we also explain what branching rules and tree heuristics are implemented in GCG.

Branching Process
Branching Rules
Branching Candidate Selection Heuristics
Diving Heuristics

As always with GCG, this algorithmic component is highly configurable. Thus, we want to present branching parameters that exist to tweak the process if you have knowledge of how you want to branch on the variables of your problem.

Branching Parameters
Parameters for Strong Branching-Based Branching Candidate Selection Heuristics

If you want to write your own plug-ins for the branching, you might want to consider our and SCIP's how-to pages on that, since some algorithmic components are implemented in and executed by SCIP only.

How to add branching rules & selection heuristics
How to add primal heuristics (deprecated)
How to add node selectors (SCIP guide)