Results on three problems on isolation of graphs

2026-02-26

AI Summary

This study investigates the F-isolation problem in graphs, focusing on its computational complexity, the influence of minimum degree on isolation numbers, and the quantitative relationship between the multi-copy structure tF and F-isolation numbers. By integrating tools from computational complexity theory, extremal graph theory, domination theory, and the Erdős–Pósa property, the work systematically resolves three fundamental challenges: it establishes that the F-isolation problem is NP-complete when F is connected; derives tight upper and lower bounds on the isolation number for graphs with sufficiently large minimum degree; and uncovers a precise quantitative connection between tF and the F-isolation number. These contributions substantially extend the theoretical foundations of graph domination.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

The graph isolation problem was introduced by Caro and Hansberg in 2015. It is a vast generalization of the classical graph domination problem and its study is expanding rapidly. In this paper, we address a number of questions that arise naturally. Let $F$ be a graph. We show that the $F$-isolating set problem is NP-complete if $F$ is connected. We investigate how the $F$-isolation number $ι(G,F)$ of a graph $G$ is affected by the minimum degree $d$ of $G$, establishing a bounded range, in terms of $d$ and the orders of $F$ and $G$, for the largest possible value of $ι(G,F)$ with $d$ sufficiently large. We also investigate how close $ι(G,tF)$ is to $ι(G,F)$, using domination and, in suitable cases, the Erdos-Posa property.

Problem

• graph isolation
• NP-completeness
• isolation number
• minimum degree
• Erdos-Posa property

Innovation

• graph isolation
• NP-completeness
• minimum degree
• domination
• Erdős–Pósa property