The 3-path isolation number of a connected n-vertex graph G, denoted by ι (G, P₃), is the size of a smallest subset D of the vertex set of G such that the closed neighbourhood ND of D in G intersects each 3-vertex path of G, meaning that no two edges of G-ND intersect. Zhang and Wu proved that ι (G, P₃) 2n/7 unless G is a 3-path or a 3-cycle or a 6-cycle. The bound is attained by infinitely many graphs having induced 6-cycles. Huang, Zhang and Jin proved that if G has no 6-cycles, or G has no induced 5-cycles and no induced 6-cycles, then ι (G, P₃) n/4 unless G is a 3-path or a 3-cycle or a 7-cycle or an 11-cycle. They asked if the bound still holds asymptotically for connected graphs having no induced 6-cycles. More precisely, taking f (n) to be the maximum value of ι (G, P₃) over all connected n-vertex graphs G having no induced 6-cycles, their question is whether ₍ f (n) n = 14. We verify this by proving that f (n) = (n+1) /4. The proof hinges on further proving that if G is such a graph and ι (G, P₃) = (n+1) /4, then ι (G-v, P₃) < ι (G, P₃) for each vertex v of G. This new idea promises to be of further use. We also prove that if the maximum degree of such a graph G is at least 5, then ι (G, P₃) n/4.
Bartolo et al. (Mon,) studied this question.