A note on the \(P_3\)-isolation number of a graph (Q6658909)

Let \(G\) be a graph, \(v\in V(G)\) and \(D\subseteq V(G)\). As usual, closed neighborhood \(N_G[v]\) contains \(v\) and all its neighbors and \(N_G[D]=\cup_{u\in D}N_G[u]\). A set \(D\) is called a \(P_3\)-isolating set if \(G-N_G[D]\) contains no path \(P_3\). The \(P_3\)-isolation number \(\iota(G,P_3)\) of \(G\) is the minimum cardinality of a \(P_3\)-isolating set.\N\NThe main result of the present contribution is that \(\iota(G,P_3)\leq \frac{n}{4}\) for a connected triangle-free and induced \(C_6\)-free graph \(G\notin\{P_3,C_3,C_6\}\) on \(n\) vertices. Moreover, this bound is sharp.