Isolation of non-triangle cycles in graphs
Peter Borg, Dayle Scicluna
TL;DR
The paper addresses the problem of bounding the F-isolation number for non-triangle cycles in graphs. It introduces the concept of F-isolating sets and focuses on the non-triangle cycle family $\mathcal{C}'$, establishing a tight bound $ι(G, \mathcal{C}') \le (m+1)/6$ for connected graphs with $m$ edges that are not 4-cycles, and fully characterizes the equality cases as pure $(m, C_4)$-special graphs or $\{C_4', C_5\}$-graphs. The result unifies and extends prior bounds for cycle families and yields as a corollary $ι(G, \{C_4\}) \le (m+1)/6$, aligning with previous work by Wei, Zhang and Zhao. The extremal graphs form infinite families, and the proof relies on a structural decomposition, induction on $m$, and a detailed analysis of residual components classified into specific special graph families, revealing the configurations that achieve equality. This contributes a precise understanding of cycle-avoidance via vertex neighborhoods and provides a framework for similar isolation-number bounds in related graph families.
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 set $D$ of vertices of $G$ 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). Let $\mathcal{C}$ be the set of cycles, and let $\mathcal{C}'$ be the set of non-triangle cycles (that is, cycles of length at least $4$). Let $G$ be a connected graph having exactly $n$ vertices and $m$ edges. The first author proved that $ι(G,\mathcal{C}) \leq n/4$ if $G$ is not a triangle. Bartolo and the authors proved that $ι(G,\{C_4\}) \leq n/5$ if $G$ is not a copy of one of nine graphs. Various authors proved that $ι(G,\mathcal{C}) \leq (m+1)/5$ if $G$ is not a triangle. We prove that $ι(G,\mathcal{C}') \leq (m+1)/6$ if $G$ is not a $4$-cycle. Zhang and Wu established this for the case where $G$ is triangle-free. Our result yields the inequality $ι(G,\{C_4\}) \leq (m+1)/6$ of Wei, Zhang and Zhao. These bounds are attained by infinitely many (non-isomorphic) graphs. The proof of our inequality hinges on also determining the graphs attaining the bound.