Complex sentence: Suppose "G" has minimum degree "d" and girth 2 "k" +1, choose arbitrarily a starting vertex "v", and as before consider the breadth first search tree rooted at "v".
Split into simpler sentences: Suppose "G" has minimum degree "d" and girth 2 "k" +1. Choose arbitrarily a starting vertex "v", and as before consider the breadth-first search tree rooted at "v".