I am studying the book "Parametrized Algorithms" and it suggests a bounded search tree algorithm for k-Vertex Cover. Basically we look at a vertex v and say :
Either v will cover it's incident edges or it has to be it's neighbors $N(V)$ so either $v$ or $N(v)$ will end up in the cover.
That's 2 choices , we build a tree with these . The total running time is summarized by this relation :
$T(k) =T(k-1) + T(k-N(v)) + O(n)$
Considering the interesting case is $N(v) \geq 2$ then $k - N(v) \leq k-2$ so we get: $$T(k) =T(k-1) + T(k-2) + O(n)$$ Now the book says that "Vertex Cover can be solved optimally in polynomial time when the maximum degree of a graph is at most 2".
"Thus, we branch only on the vertices of degree at least 3, which immediately brings us to the following upper bound on the number of leaves in a search tree:"
Since the are no vertices with degree 3 or more how do we branch on these ? What am I missing here?