Secure domination in $P_5$-free graphs

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of $G$ with the additional property that for every vertex $v \in V(G) \setminus S$, there exists a neighbor $u$ of $v$ in $S$ such that $(S \setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. The secure domination number of $G$, denoted by $\gamma_s(G)$, is the minimum cardinality of a secure dominating set of $G$. We prove that if $G$ is a $P_5$-free graph, then $\gamma_s(G) \le \frac{3}{2}\alpha(G)$, where $\alpha(G)$ denotes the independence number of $G$. We further show that if $G$ is a connected $(P_5, H)$-free graph for some $H \in \{ P_3 \cup P_1, K_2 \cup 2K_1, ~\text{paw},~ C_4\}$, then $\gamma_s(G)\le \max\{3,\alpha(G)\}$. We also show that if $G$ is a $(P_3 \cup P_2)$-free graph, then $\gamma_s(G)\le \alpha(G)+1$.