2023/05/02 by Bodlaender, Hans L., Johnson, Matthew, Martin, Barnaby +5
#Combinatorics (math.CO) #Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.01613
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. In this paper, we make significant progress towards determining the complexity of Steiner Forest on H-subgraph-free graphs. 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 S of vertices such that each component of G-S has size at most c. Using this parameter, we give two noteworthy algorithms that we later employ as subroutines. First, we prove Steiner Forest is FPT parameterized by |S| when c=1 (i.e. the vertex cover number). Second, we prove 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.