The rati of the irredundance and domination number of a graph (Q1377839)

A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is called dominating, if each vertex of \(G\) either belongs to \(D\), or is adjacent to a vertex of \(D\). A vertex \(x\) of a subset \(I\) of \(V(G)\) is called redundant in \(I\), if each vertex adjacent to \(x\) is adjacent to another vertex of \(I\). If \(I\) does not contain redundant vertices, it is called irredundant. The minimum number of vertices of a dominating (or irredundant) set in \(G\) is the domination number \(\gamma(G)\) (or irredundance number \(\text{ir}(G)\) respectively) of \(G\). The main result of the paper states that if either the cyclomatic number \(\mu(G)\) of \(G\) is at most 2, or \(G\) is a block graph, then \(2\gamma(G)\leq 3\text{ ir}(G)\). It is conjectured that \(\text{ir}(G)/\gamma(G)> 5/8\).