Isolation of squares in graphs

Abstract

Given a set $F$ of graphs, we call a copy of a graph in $F$ an $F$-graph. The $F$-isolation number of a graph G, denoted by $\iota (G,F)$, is the size of a smallest subset D of the vertex set $V\left(G\right)$ such that the closed neighbourhood of D intersects the vertex sets of the $F$-graphs contained by G (equivalently, $G-N\left[D\right]$ contains no $F$-graph). Thus, $\iota (G,\{K_1\left\}\right)$ is the domination number of G. The second author showed that if $F$ is the set of cycles and G is a connected n-vertex graph that is not a triangle, then $\iota (G,F)\le \lfloor \frac{n}{4}\rfloor$. This bound is attainable for every n and solved a problem of Caro and Hansberg. A question that arises immediately is how much smaller an upper bound can be if $F=\left\{C_k\right\}$ for some $k\ge 3$, where $C_k$ is a cycle of length k. The problem is to determine the smallest real number $c_k$ (if it exists) such that for some finite set $E_k$ of graphs, $\iota (G,\{C_k\left\}\right)\le c_k\left|V\right(G\left)\right|$ for every connected graph G that is not an $E_k$-graph. The above-mentioned result yields $c_3=\frac{1}{4}$ and $E_3=\left\{C_3\right\}$. The second author also showed that if $k\ge 5$ and $c_k$ exists, then $c_k\ge \frac{2}{2k+1}$. We prove that $c_4=\frac{1}{5}$ and determine $E_4$, which consists of three 4-vertex graphs and six 9-vertex graphs. The 9-vertex graphs in $E_4$ were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.