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arXiv · 2305.01613

Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem

Hans L. Bodlaender·Matthew Johnson·Barnaby Martin·Jelle J. Oostveen·Sukanya Pandey·Daniel Paulusma·Siani Smith·Erik Jan van Leeuwen
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Abstract

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.

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BibTeXRIS

Hans L. Bodlaender, Matthew Johnson, Barnaby Martin, Jelle J. Oostveen, Sukanya Pandey, Daniel Paulusma, Siani Smith, Erik Jan van Leeuwen. 2023-05-02. Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem. https://arxiv.org/abs/2305.01613

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