A note on the <i>P</i>₃-isolation number of a graph

For any graph G , a subset D ⊆ V ( G ) is called a P 3 -isolating set of G if G − N [ D ] contains no P 3 as a subgraph, that is, consists of isolated vertices and isolated edges only. The P 3 -isolation number of G , denoted by ι ( G, P 3 ), is the cardinality of a smallest P 3 -isolating set of G . Zhang and Wu [ Discrete Appl. Math . 304 (2021) 365–374] investigated the parameter ι ( G, P 3 ) of a graph, and they proved that if G ∉ { P 3 , C 3 , C 6 } is a connected graph of order n , then ι ( G,P 3)≤27 n . In this paper, we shall prove that if G ∉ { P 3 , C 7 , C 11 } is a connected graph of order n without triangles and induced 6-cycles, then ι ( G,P 3)≤ n 4, and the upper bound is sharp. This extends a result on ι ( T, P 3 ) of a tree T by Caro and Hansberg [ Filomat 31 (2017) 3925–3944].