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-ND contains no P₃ as a subgraph, that is, consists of isolated vertices and isolated edges only. The P₃-isolation number of G, denoted by (G, P₃), is the cardinality of a smallest P₃-isolating set of G. Zhang and Wu (2021) investigated the parameter (G, P₃) of a graph, and they proved that if G \P₃, C₃, C₆\ is a connected graph of order n, then (G, P₃) 27n. In this paper, we shall prove that if G \P₃, C₇, C₁₁\ is a connected graph of order n without triangles and induced 6-cycles, then (G, P₃) n4, and the upper bound is sharp. This extends a result on (T, P₃) of a tree T by Caro and Hansberg (2017).