The linear programme solution is an integer solution with a better objective value than the incumbent. The incumbent is replaced with this new solution;
The linear programme solution has a worse objective than the incumbent. Any nodes created from this node will also have a worse objective than the incumbent. This node is bounded by the incumbent objective;
The linear programme solution is fractional and has a better objective value than the incumbent. Further branching from this node is necessary to ensure an optimal solution is found.
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A Tie? - Cam
Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node.