Steiner Forest for $H$-Subgraph-Free Graphs

2026-02-25

AI Summary

This study investigates the computational complexity of the Steiner Forest problem on graph classes excluding an arbitrary connected graph $ H $ as a subgraph. By leveraging structural graph analysis, treewidth theory, and parameterized reductions, the paper establishes the first complete complexity dichotomy for all connected graphs $ H $: the problem is polynomial-time solvable if the graph class has bounded $ c $-deletion set number, and NP-complete otherwise. The key contribution lies in identifying two novel families of tractable graph structures, which precisely delineate the boundary of polynomial solvability for Steiner Forest on $ H $-subgraph-free graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Steiner Forest
• H-subgraph-free graphs
• computational complexity
• dichotomy
• NP-completeness

Innovation

• Steiner Forest
• H-subgraph-free graphs
• treewidth
• deletion set number
• computational dichotomy