One more example is $k$-partitioning problem, where given a set $A$ of $n$ numbers where $n$ is a multiple of $k$, the goal is to find a partition of $A$ into $k$-tuples, such that the numbers in each tuple sum up to zero.
More specifically, given a set $A$ of $m \cdot k$ elements. The goal is to partition $A$ into $m$ tuples $(a_{i1}, \dots a_{ik})$ for $i \in [m]$ each of size $k$, such that the sum of the numbers in each tuple $a_{i1} + a_{i2} \dots + a_{ik}$ is equal to zero.
The problem is hard for $k \geq 3$. However, for $k = 2$ the problem turns to find if for each value $x \in A$, whether $-x$ is also in $A$ which is doable in linear/almost-linear time. You can also define the problem on multisets and hence the positive-instance condition in case of $k = 2$ turns into $\#_A(x) = \#_A(-x)$ for all $x \in A$.
Note This problem differs from the set partitioning problem, where we are given a set $A$ and the goal is to find a partition of $A$ into $A_1$ and $A_2$ such that $\sum_{x \in A_1} x = \sum_{y \in A_2}y.$
The set partitioning problem admits a psudo-polynomial algorithm and hence is weakly NP-hard meanwhile $k$-partitioning for $k \geq 3$ is a strongly NP-hard problem. Assuming $P \neq NP$, strongly NP-hard problem are strictly harder than weakly NP-hard problem. (For instance, a pseudo polynomial algorithm for the 3-partitioning problem implies a polynomial time algorithm for the 3d-Matching problem which is an NP-hard problem).
Side note. The parameter $n$ is usually used to represent the size of the input. In Graphs it is used as the number of vertices and in sets as the number of elements. Try to avoid using $n$ to name other parameters since that might create confusion. Usually $k$ and $r$ are used for other parameters. Quite often in parameterized complexity $k$ is used as the parameter representing the size of the output (In the running time of output sensitive algorithms). For example, the independence number in the maximum independent set problem and the minimum feedback vertex number in the feedback vertex set problem.