The $k$-Vertex-Disjoint Paths Problem ($k$-$\text{DPP}$) is defined as follows:
Input: A graph $G=(V,E)$ and $k$ pairs of vertices $(s_1,t_1),\ldots,(s_k,t_k)$.
Question: Does there exist $k$-pairwise vertex-disjoint paths $P_1,\ldots,P_k$, such that $P_i$ goes from $s_i$ to $t_i$?
The problem, for general $k$, is known to be NP-complete even for planar graphs of max degree 3.
That said, $2$-$\text{DPP}$ admits a nearly linear algorithm for general undirected graphs.
Is there anything known for any higher value of $k$ (assuming fixed $k$ value)? What about $k=3$?