Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem

We study Steiner Forest on H-subgraph-free graphs, that is, graphs that do not contain some fixed graph H as a (not necessarily induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on H-subgraph-free graphs. However, in contrast to, e.g. the related Steiner Tree problem, Steiner Forest falls outside this framework. Hence, the complexity of Steiner Forest on H-subgraph-free graphs remained tantalizingly open. We make significant progress on this open problem: our main results are four novel polynomial-time algorithms for different excluded graphs H that are central to further understand its complexity. Along the way, we study the complexity of Steiner Forest for graphs with a small c-deletion set, that is, a small set X of vertices such that each component of \(G-X\) has size at most c. Using this parameter, we give two algorithms that we later employ as subroutines. First, we present a significantly faster parameterized algorithm for Steiner Forest parameterized by |X| when \(c=1\) (i.e. the vertex cover number), which by a recent result is best possible under ETH [Feldmann and Lampis, arXiv 2024]. Second, we prove that Steiner Forest is polynomial-time solvable for graphs with a 2-deletion set of size at most 2. The latter result is tight, as the problem is NP-complete for graphs with a 3-deletion set of size 2.