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

Tesshu Hanaka, Yuto Okada, Yota Otachi, Lena Volk

TL;DR

The paper investigates the parameterized complexity of Maximum Common Subgraph (MCS) and Maximum Common Induced Subgraph (MCIS) under key graph-structure parameters. It establishes fixed-parameter tractability for both problems under max-leaf number, neighborhood diversity, and twin-cover-based parameterization (induced case), and extends to cluster-vertex-deletion contexts. The approach includes twin-cover based algorithms with kernelization and weighted matching, plus ILP/IQP formulations to handle various parameter regimes, yielding a near-complete classification. The results reveal a distinct behavior between induced and non-induced variants in twin-cover settings and provide tight structural bounds that enable scalable kernelized solutions. An open question remains for MCIS parameterized by the sum of cluster-vertex-deletion numbers, suggesting a natural direction for future work.

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.

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

TL;DR

The paper investigates the parameterized complexity of Maximum Common Subgraph (MCS) and Maximum Common Induced Subgraph (MCIS) under key graph-structure parameters. It establishes fixed-parameter tractability for both problems under max-leaf number, neighborhood diversity, and twin-cover-based parameterization (induced case), and extends to cluster-vertex-deletion contexts. The approach includes twin-cover based algorithms with kernelization and weighted matching, plus ILP/IQP formulations to handle various parameter regimes, yielding a near-complete classification. The results reveal a distinct behavior between induced and non-induced variants in twin-cover settings and provide tight structural bounds that enable scalable kernelized solutions. An open question remains for MCIS parameterized by the sum of cluster-vertex-deletion numbers, suggesting a natural direction for future work.

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.

Paper Structure

This paper contains 17 sections, 11 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background of the target setting
  3. Our results
  4. Related results
  5. Preliminaries
  6. MCIS parameterized by twin cover number
  7. Guessing $\eta_{i}$ on $T$.
  8. Cleaning up $G_{i}$.
  9. Finding a maximum solution under the guesses.
  10. MCIS parameterized by cluster vertex deletion number
  11. MCS and MCIS parameterized by max-leaf number
  12. MCS and MCIS parameterized by neighborhood diversity
  13. Conclusion
  14. Tight bound on the number of non-degree-2 vertices
  15. Initializing $T$.
  16. ...and 2 more sections

Key Result

theorem thmcountertheorem

Maximum Common Induced Subgraph is fixed-parameter tractable parameterized by $\mathop{\mathrm{\mathsf{tc}}}\nolimits(G_{1})+\mathop{\mathrm{\mathsf{tc}}}\nolimits(G_{2})$.

Figures (1)

  • Figure 1: The complexity of MCS (left) and MCIS (right) when a structural parameter of both input graphs is bounded. The normal rectangles and the rounded rectangles represent paraNP-complete cases and fixed-parameter tractable cases, respectively. The results marked with $\star$ are shown in this paper and the ones with $\circ$ are corollaries of the observations in \ref{['sec:background']}. A connection between two parameters means that the one above is upper-bounded by a function of the one below (e.g., $\mathop{\mathrm{\mathsf{{tw}}}}\nolimits(G) \le \mathop{\mathrm{\mathsf{pw}}}\nolimits(G)$).

Theorems & Definitions (24)

  • theorem thmcountertheorem
  • proof
  • proof : \ref{['clm:tc-only-for-T']}
  • proof : \ref{['clm:tc-clean-up']}
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • ...and 14 more