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Some Stability Results on Graphs

Angshuman R. Goswami, Mahmood K. Shihab

TL;DR

Extends Hyers-Ulam stability to graphs by defining approximate monotonicity, subadditivity, and convexity on the subgraph poset and proving that any graph satisfying these approximate properties is within a bounded distance of an exact graph with the property. The paper provides constructive procedures to obtain a monotone minorant, a subadditive minorant, and a convex minorant with bounds: $||w-\tilde w|| \le \varepsilon/2$, $||w-\overline w|| \le \varepsilon$, and $||w-\widehat w|| \le \varepsilon/2$, respectively. It further proves the corresponding converse statements, discusses generality to infinite vertex/edge sets, and outlines future directions such as specialized graph classes and multidimensional error models.

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.

Some Stability Results on Graphs

TL;DR

Extends Hyers-Ulam stability to graphs by defining approximate monotonicity, subadditivity, and convexity on the subgraph poset and proving that any graph satisfying these approximate properties is within a bounded distance of an exact graph with the property. The paper provides constructive procedures to obtain a monotone minorant, a subadditive minorant, and a convex minorant with bounds: , , and , respectively. It further proves the corresponding converse statements, discusses generality to infinite vertex/edge sets, and outlines future directions such as specialized graph classes and multidimensional error models.

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.
Paper Structure (1 section, 3 theorems, 40 equations)

This paper contains 1 section, 3 theorems, 40 equations.

Table of Contents

  1. Main Results

Key Result

Proposition 1

Let $G(V,E,w)$ be an approximately monotone graph, then there exists a monotone(increasing) graph $G(V,E,\widetilde{w})$ such that $\|w-\widetilde{w}\|\leq \varepsilon/2$ holds. Conversely if $G(V,E,\widetilde{w})$ is a monotone graph satisfying the inequality $\|w-\widetilde{w}\|\leq \varepsilon/2$

Theorems & Definitions (6)

  • Proposition
  • proof
  • Theorem
  • proof
  • Theorem
  • proof