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 γ_s(G), is the minimum cardinality of a secure dominating set of G . We prove that if G is a P₅-free graph, then γ_s(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 γ_s(G)≤max⁡{3,α(G)}. We also show that if G is a (P₃∪P₂)-free graph, then γ_s(G)≤α(G)+1.