Algorithmic aspects of {Pk}-isolation in graphs and extremal graphs for a {P3}-isolation bound

Abstract

A subset S of the vertex set of a graph G is an $F$-isolating set of G if $G-N\left[S\right]$ does not contain a copy of a member of $F$ as a subgraph, where $F$ is a family of connected graphs and $N\left[S\right]$ is the closed neighborhood of S. The $F$-isolation number of G is the minimum cardinality of an $F$-isolating set of G, denoted by $\iota (G,F)$. Given a graph G, $\left\{P_k\right\}$-ISOLATION asks for the size of a smallest $\left\{P_k\right\}$-isolating set of G for a fixed positive integer k, where $P_k$ is a path of order k. In this paper, we first show that the decision version of $\left\{P_k\right\}$-ISOLATION is NP-complete for chordal graphs and planar graphs. Secondly, we propose a linear time algorithm to compute a smallest $\left\{P_k\right\}$-isolating set of a tree. Finally, we solve the problem of characterizing the connected graphs G with $\iota (G,\{P_3\left\}\right)=\frac{2}{7}\left|V\right(G\left)\right|$, proposed by Zhang and Wu [Discrete Appl. Math. 304 (2021) 365-374].