I am trying to solve the following exercise from this book:Show that CLIQUE PROBLEM, parameterized by the solution size $k$, is  Fixed-parameter tractable (FTP) on $r$-regular graphs for every  fixed integer $r$.Here, the CLIQUE PROBLEM is given a instance $(G, k)$, decide whether $G$ has a clique of size $k$ or not.First of all, for an instance $(G, k)$,  if $k > r+1$, then the answer is NO, because each vertex is connected with exactly $r$ elements, the maximum size of a clique is $r + 1$ (vertex plus $r$ neighbours). So, we can assume that $k \le r+1$.Let $N(v)$ be the set of neighbours of $v$.I thought of that simple algorithm.... for each vertex $v \in V(G)$........ check if for any subset $X \subset N(v)$, such that $|X| = k - 1$, $X \cup \{v\}$ is a clique.Since there is only $\binom r k$ such subsets $X$ for each vertex and we take time polynomial in $k$ to check if $X \cup \{v\}$ is a clique, then, this algorithm is already a FTP and is of the form $\left( k^{O(1)}\binom{r}{k} \right)n$.If everything is right, them I have solved the exercise. However, the next thing I have to do in the exercise, is to show that this problem is also a FTP considering the parameter $k + r$ (so, $r$ is no longer seen as a constant), and the same algorithm works in this case. Since I was expecting to face a harder exercise in this case of $k + r$, I started to think my solution is not right.So, what is wrong?
Parameterized vertex cover on $r$-regular graphs
complexity theory;graphs;clique