Improved secure-domination bound for connected P5-free graphs

Determine whether the bound \(\gamma_s(G)\le \frac{3}{2}\alpha(G)\) can be improved for every connected \(P_5\)-free graph \(G\) satisfying \(\alpha(G)\ge 3\).

Background

The paper proves that every $P_5$-free graph $G$ satisfies $\gamma_s(G)\le \frac{3}{2}\alpha(G)$, and notes that this bound is optimal in general because disjoint unions of copies of $C_5$ attain equality. Since those extremal examples are disconnected and have independence number 2 per component, the authors explicitly ask whether connectivity together with $\alpha(G)\ge 3$ permits a strictly stronger upper bound.