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. In contrast to the related Steiner Tree problem, Steiner Forest falls outside a recent framework that completely characterizes the complexity of many problems on H-subgraph-free graphs. Hence, the complexity of Steiner Forest on H-subgraph-free graphs remained open. Our main results are four polynomial-time algorithms for different excluded graphs H that are central to further understand its complexity. We also 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 connected component of $G-X$ has size at most c. For this parameter, we give two algorithms that we later employ as subroutines (including a faster algorithm when $c=1$, that is, the vertex cover number) and exhibit a dichotomy theorem.