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 \u\) \v\ 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) 32α (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) \3, α (G) \. We also show that if G is a (P₃ P₂) -free graph, then γₛ (G) α (G) +1.