Domination and irredundance in graphs with restricted blocks (Q2713990)

Let \(G=(V,E)\) be a graph, and let \(N[V']\) be the closed neighborhood of \(V'\subseteq V\). The domination number \(\gamma(G)\) is the smallest \(|V'|\) such that \(N[V']=V\). If \(v\in V'\) then \(v\) is irreducible with respect to \(V'\) in \(G\) provided that \(PN(v)=N[v]\setminus N[V'\setminus\{v\}]\neq\varnothing\). A vertex set \(V'\subseteq V\) is irreducible in \(G\) if each \(v\in V'\) is irreducible with respect to \(V'\) in \(G\). The irredundance number \(\text{ir}(G)\) of \(G\) is defined as the minimal \(|V'|\) such that \(V'\) is a maximal (by inclusion) irreducible set in \(G\). The article provides upper bounds for \(\frac{\gamma(G)}{\text{ir}(G)}\) if each block \(B\) of \(G\) has the property that all the cut-vertices of \(G\) in \(B\) form a clique and, furthermore, either \(G\) is claw-free or each \(B\) is the line graph of a triangle-free graph.