Revising Johnson's table for the 21st century

TL;DR

This paper revises Johnson's classic table on graph restrictions and computational complexity, providing a complete classification for the Steiner Tree problem on specific graph classes using parameterized complexity.

Contribution

It offers a full NP-completeness dichotomy for the Steiner Tree problem on Undirected Path graphs, updating Johnson's 35-year-old classification table.

Findings

Steiner Tree is NP-complete on Undirected Path graphs.

Provides a detailed classification for 30 graph classes.

Updates Johnson's table with modern complexity insights.

Abstract

What does it mean today to study a problem from a computational point of view? We focus on parameterized complexity and on Column 16 "Graph Restrictions and Their Effect" of D. S. Johnson's Ongoing guide, where several puzzles were proposed in a summary table with 30 graph classes as rows and 11 problems as columns. Several of the 330 entries remain unclassified into Polynomial or NP-complete after 35 years. We provide a full dichotomy for the Steiner Tree column by proving that the problem is NP-complete when restricted to Undirected Path graphs. We revise Johnson's summary table according to the granularity provided by the parameterized complexity for NP-complete problems.