9.4 In Greek mythology, the "many-headed serpent" god is composed of some heads and necks, with each neck connecting two heads. With each strike of a sword, one can sever all the necks connected to a certain head $A$. However, head $A$ immediately grows new necks connecting to all the heads it was not previously connected to (each head is connected by only one neck). Only by severing the "many-headed serpent" into two mutually disconnected parts can it be defeated. Try to find the smallest natural number $N$ such that for any "many-headed serpent" god with 100 necks, it can be defeated with no more than $N$ strikes.
9.410 .

We will reformulate the problem using graph theory terminology, with heads as vertices, necks as edges, and a strike that cuts the necks connected to head $A$ as a "reversal" of vertex $A$. It is easy to see that if a vertex $X$ has a degree no greater than 10, then it is sufficient to perform a "reversal" on all vertices adjacent to $X$ to make vertex $X$ "isolated". If a vertex $X$ is not adjacent to at most $n (n \leqslant 9)$ vertices, then it is sufficient to first perform a "reversal" on $X$, and then perform a "reversal" on each of these $n$ vertices to make vertex $X$ "isolated".

If each vertex has at least 11 adjacent vertices and at least 10 non-adjacent vertices, then there must be at least 22 vertices. Thus, the number of edges (necks) is no less than $22 \times 11 > 100$, which is not within the scope of consideration.
We provide an example to show that 9 strikes may not be sufficient.
Assume there are two groups of 10 heads each, with each head in one group connected to each head in the other group, resulting in exactly 100 necks.
If 9 strikes are made, then there is at least one head in each group that is not struck, denoted as $A$ and $B$. For the remaining 18 heads, by the problem's conditions, any head $C$ is connected to exactly one of $A$ or $B$ at any time (during the 9 strikes), and not to the other. Since $A$ and $B$ are connected, the "many-headed snake" remains connected.