On the extremal graphs for degenerate subsets, dynamic monopolies, and partial incentives

TL;DR

This paper characterizes extremal graphs related to bounds on degenerate subsets, dynamic monopolies, and partial incentives, extending classical results and providing simplified proofs for these bounds.

Contribution

It generalizes known bounds to broader graph classes, characterizes extremal graphs for these bounds, and offers a simplified proof for partial incentives.

Findings

Characterization of extremal graphs for Ackerman, Ben-Zwi, and Wolfovitz's generalization.

Simplified proof of the bound on partial incentives.

Identification of extremal graphs for partial incentives.

Abstract

The famous lower bound $\alpha(G)\geq \sum_{u\in V(G)}\frac{1}{d_G(u)+1}$ on the independence number $\alpha(G)$ of a graph $G$ due to Caro and Wei is known to be tight if and only if the components of $G$ are cliques, and has been generalized several times in the context of large degenerate subsets and small dynamic monopolies. We characterize the extremal graphs for a generalization due to Ackerman, Ben-Zwi, and Wolfovitz. Furthermore, we give a simple proof of a related bound concerning partial incentives due to Cordasco, Gargano, Rescigno, and Vaccaro, and also characterize the corresponding extremal graphs.