Isolation of squares in graphs
Karl Bartolo, Peter Borg, Dayle Scicluna
TL;DR
The paper investigates the $ι(G, {C_k})$ isolation problem for cycles, aiming to bound it by a fixed fraction of the order of $G$ and to identify exceptional graphs where the bound is not tight. For the cycle case $k=4$, it establishes a sharp bound $ι(G,{C_4}) \le floor(|V(G)|/5)$ and determines the exceptional family $E_4$, which includes three 4-vertex graphs and six 9-vertex graphs (the latter found by computer search). Equality is achieved by the construction $B_{n, C_4}$, and the six 9-vertex graphs are explicitly enumerated, illustrating a robust, computable approach to extremal instances. The proof combines subcubic-case analysis with a general inductive framework that removes a closed neighborhood and analyzes attached components, offering a scalable method that may extend to other $k$ and yield further exact extremal results. This advances the understanding of cycle-isolation and connects to broader domination-type parameters and previous cycle-based results.
Abstract
Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $ι(G,\mathcal{F})$, is the size of a smallest subset $D$ of the vertex set $V(G)$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G - N[D]$ contains no $\mathcal{F}$-graph). Thus, $ι(G,\{K_1\})$ is the domination number of $G$. The second author showed that if $\mathcal{F}$ is the set of cycles and $G$ is a connected $n$-vertex graph that is not a triangle, then $ι(G,\mathcal{F}) \leq \left \lfloor \frac{n}{4} \right \rfloor$. 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 $\mathcal{F} = \{C_k\}$ for some $k \geq 3$, 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 $\mathcal{E}_k$ of graphs, $ι(G, \{C_k\}) \leq c_k |V(G)|$ for every connected graph $G$ that is not an $\mathcal{E}_k$-graph. The above-mentioned result yields $c_3 = \frac{1}{4}$ and $\mathcal{E}_3 = \{C_3\}$. The second author also showed that if $k \geq 5$ and $c_k$ exists, then $c_k \geq \frac{2}{2k + 1}$. We prove that $c_4 = \frac{1}{5}$ and determine $\mathcal{E}_4$, which consists of three $4$-vertex graphs and six $9$-vertex graphs. The $9$-vertex graphs in $\mathcal{E}_4$ were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.