Isolation of non-triangle cycles in graphs

2025-10-09

AI Summary

This paper studies the *non-triangular cycle isolation problem* on graphs: find a minimum vertex set (D) such that its closed neighborhood intersects every cycle of length at least four (equivalently, deleting this closed neighborhood eliminates all non-triangular cycles). We establish, for the first time, that in any (C_4)-free graph, the non-triangular cycle isolation number is at most ((m+1)/6), where (m) denotes the number of edges; this bound is tight and attained by infinitely many pairwise non-isomorphic extremal graphs. Moreover, we fully characterize the structure of all extremal graphs. Our approach integrates extremal graph theory, fine-grained structural analysis, and constructive case classification. The result generalizes and strengthens several prior bounds in the literature and yields the optimal (C_4) isolation number as a corollary.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 $iota(G, mathcal{F})$, 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 $iota(G,mathcal{C}) leq n/4$ if $G$ is not a triangle. Bartolo and the authors proved that $iota(G,{C_4}) leq n/5$ if $G$ is not a copy of one of nine graphs. Various authors proved that $iota(G,mathcal{C}) leq (m+1)/5$ if $G$ is not a triangle. We prove that $iota(G,mathcal{C}') leq (m+1)/6$ 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 $iota(G,{C_4}) leq (m+1)/6$ 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.

Problem

• Determining isolation number bounds for non-triangle cycles in graphs
• Establishing sharp upper bounds on vertex isolation sets
• Characterizing extremal graphs achieving the isolation number bound

Innovation

• Isolates non-triangle cycles using vertex sets
• Bounds isolation number by edge count ratio
• Determines extremal graphs achieving the bound