An exact characterization of tractable demand patterns for maximum disjoint path problems

We study the following general disjoint paths problem: given a supply graph $G$, a set $T\subseteq V(G)$ of terminals, a demand graph $H$ on the vertices $T$, and an integer $k$, the task is to find a set of $k$ pairwise vertex-disjoint valid paths, where we say that a path of the supply graph $G$ is valid if its endpoints are in $T$ and adjacent in the demand graph $H$. For a class $\mathcal{H}$ of graphs, we denote by $\mathcal{H}$-Maximum Disjoint Paths the restriction of this problem when the demand graph $H$ is assumed to be a member of $\mathcal{H}$. We study the fixed-parameter tractability of this family of problems, parameterized by $k$. Our main result is a complete characterization of the fixed-parameter tractable cases of $\mathcal{H}$-Maximum Disjoint Paths for every hereditary class $\mathcal{H}$ of graphs: it turns out that complexity depends on the existence of large induced matchings and large induced skew bicliques in the demand graph $H$ (a skew biclique is a bipartite graph on vertices $a_1$, $\dots$, $a_n$, $b_1$, $\dots$, $b_n$ with $a_i$ and $b_j$ being adjacent if and only if $i\le j$). Specifically, we prove the following classification for every hereditary class $\mathcal{H}$.

1. If $\mathcal{H}$ does not contain every matching and does not contain every skew biclique, then $\mathcal{H}$-Maximum Disjoint Paths is FPT.
2. If $\mathcal{H}$ does not contain every matching, but contains every skew biclique, then $\mathcal{H}$-Maximum Disjoint Paths is W[1]-hard, admits an FPT approximation, and the valid paths satisfy an analog of the Erd\H{o}s-P'osa property.
3. If $\mathcal{H}$ contains every matching, then $\mathcal{H}$-Maximum Disjoint Paths is W[1]-hard and the valid paths do not satisfy the analog of the Erd\H{o}s-P'osa property.