Extension of the secure-domination bound to Pt-free graphs

Determine whether every \(P_t\)-free graph \(G\), for each integer \(t\ge 6\), satisfies the bound \(\gamma_s(G)\le \frac{3}{2}\alpha(G)\).

Background

The paper establishes the bound $\gamma_s(G)\le \frac{3}{2}\alpha(G)$ for $P_5$-free graphs. The authors explicitly leave unresolved whether the same relationship continues to hold for the broader classes of $P_t$-free graphs when the forbidden induced path has length parameter $t\ge 6$.