Ramsey-type results on parameters related to domination

TL;DR

This paper characterizes graph families where certain domination-related parameters are bounded in connected graphs, extending previous results to include parameters along the domination chain and related concepts.

Contribution

It completes the characterization of graph families for which various domination parameters are bounded in connected graphs, including new parameters related to irredundance and saturation.

Findings

Characterized families where ir(G), i(G), Γ(G), IR(G) are bounded in connected graphs.

Extended characterization to parameters related to domination number, such as OIR(G), IS(G), IRS(G).

Provides a comprehensive understanding of boundedness conditions for domination parameters in graph theory.

Abstract

The following inequality chain $$ ir(G)\le \gamma(G)\le i(G)\le \alpha(G) \le \Gamma(G) \le I\!R(G)$$ is known as a domination chain, where $ir(G), \gamma(G), i(G), \alpha(G), \Gamma(G)$, and $I\!R(G)$ are the lower irredundance number, the domination number, the independence domination number, the independence number, the upper domination number and the upper irredundance number of $G$, respectively. The Ramsey-type problem seeks to characterize the family $\mathcal H$ of graphs such that every $\mathcal H$-free graph $G$ has a bounded parameter $\mu$. The classical Ramsey's theorem states that every $\{K_n, E_n\}$-free graph has a bounded number of vertices. Furuya (Discrete Math.Theor 2018) characterized $\mathcal H$ such that every connected $\mathcal H$-free graph $G$ has a bounded domination number. The characterization of the graph family $\mathcal H$ for which every connected $\mathcal H$-free graph $G$ has a bounded independence number was due to Choi, Furuya, Kim, Park~(Discrete math. 2020) and Chiba, Furuya (Electron. J. Combin., 2022). In this paper, we further characterize $\mathcal H$ such that every connected $\mathcal H$-free graph $G$ has bounded $\mu(G)$ for $\mu$ belonging to the set $\{ir(G), i(G), \Gamma(G), \text{IR}(G)\}$. This completes the characterization of $\mathcal H$ for which every connected $\mathcal H$-free graph $G$ has bounded $\mu(G)$ for $\mu(G)$ along the domination chain. Additionally, we characterize $\mathcal H$ such that every connected $\mathcal H$-free graph $G$ has bounded $\mu(G)$ for $\mu$ related to the domination number. Specifically, we consider the parameters $O\!I\!R(G)$, $I\!S(G)$, or $I\!R\!S(G)\}$, where $O\!I\!R(G)$, $I\!S(G)$, and $I\!R\!S(G)$ are the open irredundance number, the independence saturation number, and the irredundance saturation number of graph $G$, respectively.