Parameterized complexity from P to NP-hard and back again

I'm looking for examples of problems parametrized by a number $k \in \mathbb{N}$, where the problem's hardness is non-monotonic in $k$. Most problems (in my experience) have a single phase transition, for example $k$-SAT has a single phase transition from $k \in \{1,2\}$ (where the problem is in P) to $k \ge 3$ (where the problem is NP-complete). I'm interested in problems where there are phase transitions in both directions (from easy to hard and vice-versa) as $k$ increases. My question is somewhat similar to the one asked at Hardness Jumps in Computational Complexity, and in fact some of the responses there are relevant to my question. Examples I am aware of: $k$-colorability of planar graphs: In P except when $k=3$, where it is NP-complete. Steiner tree with $k$ terminals: In P when $k=2$ (collapses to shortest $s$-$t$ path) and when $k=n$ (collapses to MST), but NP-hard "in between". I don't know if these phase transitions are sharp (e.g., P for $k_0$ but NP-hard for $k_0+1$). Also the transitions of $k$ depend on the size of input instance, unlike my other examples. Counting satisfying assignments of a planar formula modulo $n$: In P when $n$ is a Mersenne prime number $n=2^k-1$, and #P-complete for most(?)/ all other values of $n$ (from Aaron Sterling in this thread ). Lots of phase transitions! Induced subgraph detection: The problem is not parametrized by an integer but a graph. There exist graphs $H_1 \subseteq H_2 \subseteq H_3$ (where $\subseteq$ denotes a certain kind of subgraph relation), for which determining whether $H_i \subseteq G$ for a given graph $G$ is in P for $i\in \{1,3\}$ but NP-complete for $i=2$. (from Hsien-Chih Chang in the same thread ).