Parameterized Complexity of Finding a Maximum Common Vertex Subgraph Without Isolated Vertices

2026-02-11

AI Summary

This work investigates the parameterized complexity of the Maximum Common Induced Subgraph without Isolated Vertices problem—specifically, finding a largest common induced subgraph with at least $h$ vertices and no isolated vertices in two given graphs. As this problem is NP-hard, the study presents the first fixed-parameter tractable (FPT) algorithm parameterized by $h$. It further provides a systematic analysis of the problem’s behavior under classical structural graph parameters, including vertex cover number, treewidth, and pathwidth. Through a combination of parameterized reductions and algorithmic design, the project establishes a complete dichotomy across multiple parameterizations, fully characterizing the parameterized complexity landscape of this problem and filling a theoretical gap concerning the isolated-vertex-free constraint in common subgraph problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In this paper, we study the Maximum Common Vertex Subgraph problem: Given two input graphs $G_1,G_2$ and a non-negative integer $h$, is there a common subgraph $H$ on at least $h$ vertices such that there is no isolated vertex in $H$. In other words, each connected component of $H$ has at least $2$ vertices. This problem naturally arises in graph theory along with other variants of the well-studied Maximum Common Subgraph problem and also has applications in computational social choice. We show that this problem is NP-hard and provide an FPT algorithm when parameterized by $h$. Next, we conduct a study of the problem on common structural parameters like vertex cover number, maximum degree, treedepth, pathwidth and treewidth of one or both input graphs. We derive a complete dichotomy of parameterized results for our problem with respect to individual parameterizations as well as combinations of parameterizations from the above structural parameters. This provides us with a deep insight into the complexity theoretic and parameterized landscape of this problem.

Problem

• Maximum Common Vertex Subgraph
• Isolated Vertices
• Parameterized Complexity
• Graph Theory
• NP-hard

Innovation

• Maximum Common Vertex Subgraph
• Fixed-Parameter Tractability
• Parameterized Complexity
• Structural Graph Parameters
• Dichotomy