Secure domination in P₅-free graphs

A dominating set of a graph G is a set S ⊆ V(G) such that every vertex in V(G) 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 ∈ V(G) S, there exists a neighbor u of v in S such that (S \) ∪ \ is a dominating set of G. The secure domination number of G, denoted by γₛ(G), is the minimum cardinality of a secure dominating set of G. We prove that if G is a P₅-free graph, then γₛ(G) ≤ 3/2α(G), where $α(G)$ denotes the independence number of G. We further show that if G is a connected (P₅, H)-free graph for some H ∈ \ P₃ ∪ P₁, K₂ ∪ 2K₁, ~paw,~ C₄\, then γₛ(G)≤ max\3,α(G)\. We also show that if G is a (P₃ ∪ P₂)-free graph, then γₛ(G)≤ α(G)+1.