A note on the P₃-isolation number of a graph

For any graph G, a subset D ⊆ V(G) is called a P₃-isolating set of G if $G-N[D]$ contains no P₃ as a subgraph, that is, consists of isolated vertices and isolated edges only. The P₃-isolation number of G, denoted by ι(G,P₃), is the cardinality of a smallest P₃-isolating set of G. Zhang and Wu (2021) investigated the parameter ι(G,P₃) of a graph, and they proved that if G ∉ ₃,C₃,C₆\ is a connected graph of order n, then ι(G,P₃) ≤ 2/7n. In this paper, we shall prove that if G ∉ ₃,C₇,C₁₁\ is a connected graph of order n without triangles and induced 6-cycles, then ι(G,P₃) ≤ n/4, and the upper bound is sharp. This extends a result on ι(T,P₃) of a tree T by Caro and Hansberg (2017).