Robustness of graph properties

TL;DR

This survey explores different measures of robustness in graph properties, discussing how graphs can strongly satisfy properties and illustrating these concepts with examples.

Contribution

It provides a comprehensive overview of measures for robustness in graph properties, highlighting their applications and significance.

Findings

Various measures of robustness are applicable to graph properties.

Robustness measures can strengthen classical graph theorems.

Examples illustrate practical applications of robustness concepts.

Abstract

A typical result in graph theory says that a graph $G$, satisfying certain conditions, has some property $\cal P$. Once such a theorem is established, it is natural to ask how strongly $G$ satisfies $\cal P$. Can one strengthen the result by showing that $G$ possesses $\cal P$ in a robust way? What measures of robustness can one utilize? In this survey, we discuss various measures that can be used to study robustness of graph properties, illustrating them with examples.