Multiplicative and mining property for stability numbers of graphs
Metrose Metsidik, Lixiao Xiao
TL;DR
This work defines and analyzes $f$-vertex and $f$-edge stability numbers for finite graphs, focusing on invariants $f$ that exhibit multiplicative or mining behavior under disjoint unions. It derives general bounds and componentwise relations by leveraging the structural decomposition of graphs into induced subgraphs and disjoint unions, and connects these stability notions to classical parameters such as the maximum degree $\Delta(G)$, minimum degree $\delta(G)$, and total chromatic stability. Key contributions include a bound $vs_f(G)\le \delta(G)+1$ for non-complete graphs when $f(K_1)\neq 1$, exact expressions for $vs_f(G)$ on disjoint unions, and analogous results for edge stability $es_f(G)$, including the link to the covering number $\beta_f'(G)$. The results provide a general framework to analyze how robust a broad class of graph invariants is to vertex/edge deletions, with implications for component-wise analysis and coloring stability.
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.