Ramsey-type results on parameters related to domination

The inequality chain \(ir(G)\le \gamma(G)\le i(G)\le \alpha(G) \le \Gamma(G) \le I\!R(G)\) is known as the 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 following parameters of \(G\): open irredundance number \(O\!I\!R(G)\), independence saturation number \(I\!S(G)\) and irredundance saturation number \(I\!R\!S(G)\).