Some Stability Results on Graphs

TL;DR

This paper establishes stability results for certain classes of graphs, showing that approximate properties imply the existence of exact graphs with similar structure and controlled weight differences.

Contribution

It introduces Hyers-Ulam-type stability results for monotone, subadditive, and convex graphs, connecting approximate and exact structural properties.

Findings

Existence of exact graphs with the same vertices and edges under approximate conditions

Weight differences between approximate and exact graphs are bounded by the error

Stability results apply to monotone, subadditive, and convex graphs

Abstract

The primary objective of this paper is to introduce Hyers-Ulam-type stability results for monotone, subadditive, and convex graphs. We consider their standard definitions in an approximate sense and demonstrate the existence of a corresponding graph with the same vertex and edge sets bearing the exact ideal structural property. We prove that the weight difference on the two graphs depends on the associated error and does not vary significantly.