Finding a Maximum Common (Induced) Subgraph: Structural Parameters Revisited

TL;DR

This paper explores the parameterized complexity of maximum common subgraph problems, identifying new fixed-parameter tractable cases based on structural graph parameters, thus nearly completing their complexity classification.

Contribution

It establishes fixed-parameter tractability for both problems under various parameters, including max-leaf number, neighborhood diversity, and twin cover number, clarifying their complexity landscape.

Findings

Fixed-parameter tractability for max-leaf number and neighborhood diversity.

Induced problem is FPT when parameterized by twin cover number.

Almost complete complexity classification of the problems based on structural parameters.

Abstract

We study the parameterized complexity of the problems of finding a maximum common (induced) subgraph of two given graphs. Since these problems generalize several NP-complete problems, they are intractable even when parameterized by strongly restricted structural parameters. Our contribution in this paper is to sharply complement the hardness of the problems by showing fixed-parameter tractable cases: both induced and non-induced problems parameterized by max-leaf number and by neighborhood diversity, and the induced problem parameterized by twin cover number. These results almost completely determine the complexity of the problems with respect to well-studied structural parameters. Also, the result on the twin cover number presents a rather rare example where the induced and non-induced cases have different complexity.