Multiplicative and mining property for stability numbers of graphs

TL;DR

This paper investigates stability numbers based on multiplicative and mining properties in graphs, providing bounds and relations between stability numbers of graphs and their components.

Contribution

It introduces bounds and relations for $f$-vertex and $f$-edge stability numbers for graphs with multiplicative and mining invariants, expanding understanding of graph stability measures.

Findings

Established bounds for $f$-vertex and $f$-edge stability numbers.

Analyzed relations between stability numbers of graphs and their components.

Extended stability concepts to multiplicative and mining invariants.

Abstract

$f$-vertex stability number $vs_f(G)=\min\{|X|: X\subseteq V(G) \enspace \text{and} \enspace f(G-X)\neq f(G)\}$, and $f$-edge stability number is defined similarly by setting $X\subseteq E(G)$. In this paper, for multiplicative and mining invariant $f$, we give some general bounds for $f$-vertex/edge stability numbers of graphs and some results about the relations between the $f$-vertex/edge stability numbers of graphs and their components.