Given a graph class 𝒢, the limiting density of 𝒢 is defined as δ(𝒢) = lim_{n → ∞} ex(𝒢,n)/n where ex(𝒢,n) is the maximum number of edges of a graph in 𝒢 on n vertices. The limiting density δ(𝒢) is known to be a rational number when 𝒢 is a minor-closed graph class. For every δ ∈ [0,3/2), we prove that the set of ⊆-minimal minor-closed graph classes with densities > δ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs 𝒵, of total size n, either outputs the value of δ(excl(𝒵)) or reports that δ(excl(𝒵)) ≥ 3/2, where excl(𝒵) is the class of graphs excluding the graphs in 𝒵 as minors. The algorithm runs in 2^{poly(n)} time.

We study the problem of reconfiguring one minimum $s$-$t$-separator $A$ into another minimum $s$-$t$-separator $B$ in some $n$-vertex graph $G$ containing two non-adjacent vertices $s$ and $t$. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming $A$ into $B$. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most $\ell$ jumps can transform $A$ into $B$ is an $\textsf{NP}$-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size $k$ of the minimum \stseps and when parameterized by the number of jumps $\ell$. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complete the picture by designing a kernel with $\mathcal{O}(\ell^2)$ vertices and edges for the length $\ell$ of the sequence as a parameter.