Cutting the undersea cables near Yemen isn't going to fragmentate or break the internet, at most it's going to make it slower to connect between India (and East Asia) and Europe.

The most important "Internet backbones" are the cables connecting Europe and North America, the Atlantic Backbone, and the cables connecting East Asia and North America, the Pacific Backbone.
If anything would happen to those two, then you can start to shit bricks for real.