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).
One field with lots of non-monotonicity of problem complexity is property testing. Let $\mathcal{G}_n$ be the set of all $n$-vertex graphs, and call $P \subseteq \mathcal{G}_n$ a graph property. A generic problem is to determine whether a graph $G$ has property $P$ (i.e. $G \in P$) or is `far' from having property $P$ in some sense. Depending on what $P$ is, and what kind of query access you have to the graph, the problem can be quite difficult.

But it is easy to see that the problem is non-monotone, in that if we have $S \subset P \subset T$, the fact that $P$ is easily testable does not imply either that $S$ is easily testable or that $T$ is.  

To see this, it is enough to observe that $P = \mathcal{G}_n$ and $P = \emptyset$ are both trivially testable, but that for some properties, there exist strong lower bounds.