Steiner Forest for $H$-Subgraph-Free Graphs

TL;DR

The results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2 and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

Abstract

Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and combinatorial results: Algorithms: We identify two new classes of graph-theoretical structures that make it possible to solve Steiner Forest in polynomial time. Roughly speaking, our algorithms handle the following cases: (1) a set $X$ of vertices of bounded size that are pairwise connected by subgraphs of treewidth $2$ or bounded size, possibly together with an independent set of arbitrary size that is connected to $X$ in an arbitrary way; (2) a set $X$ of vertices of arbitrary size that are pairwise connected in a cyclic manner by subgraphs of treewidth $2$ or bounded size. Hardness results: We show that Steiner Forest remains NP-complete for graphs with 2-deletion set number $3$. (The $c$-deletion set number is the size of a smallest cutset $S$ such that every component of $G-S$ has at most $c$ vertices.) Combinatorial results: To establish the dichotomy, we perform a delicate graph-theoretic analysis showing that if $H$ is a path or a subdivided claw, then excluding $H$ as a subgraph either yields one of the two algorithmically favourable structures described above, or yields a graph class for which NP-completeness of Steiner Forest follows from either our new hardness result or a previously known one. Along the way to classifying the hardness for excluded subgraphs, we establish a dichotomy for graphs with $c$-deletion set number at most $k$. Specifically, our results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2$ and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

Figures (3)

• Figure 1: Left: A lemon. Right: A citrus bush flowering from a $4$-vertex path $u_1u_2u_3u_4$, where each citrus is has only juicy wedges. Note that $\{u_1,u_2,u_3,u_4\}$ is the stem set.
• Figure 2: Forbidden jumps due to occurrences of $S_{2,a,b}$ in the proof of Lemma \ref{['l-s2ab']}.
• Figure 3: The graph $G$ from the proof of Theorem \ref{['t-newhard']}.

Theorems & Definitions (62)

• Theorem 1
• Theorem 2
• Theorem 3: BBJPPL21
• Theorem 4: BHM11
• Theorem 5: BJMOPPSV25
• Lemma 6: BJMOPPSV25
• Lemma 6
• Definition 6
• Lemma 6
• Definition 6