Solution to a problem on isolation of $3$-vertex paths

2025-06-23

AI Summary

This paper addresses an open problem by Huang–Zhang–Jin concerning the asymptotic upper bound of the $P_3$-isolation number $iota(G, P_3)$—the minimum size of a vertex set whose closed neighborhood intersects every 3-vertex path—in connected $n$-vertex graphs $G$ containing no induced 6-cycle. The central question is whether $limsup_{n oinfty} f(n)/n = 1/4$, where $f(n) = max iota(G, P_3)$ over all such graphs. Method: Combining extremal graph analysis, structural induction, and neighborhood covering techniques, the authors classify critical configurations according to vertex degrees and characterize extremal structures. Contribution/Results: They prove $iota(G, P_3) le lfloor (n+1)/4 floor$ for all such $G$, with the stronger bound $iota(G, P_3) le n/4$ when $Delta(G) ge 5$. Moreover, extremal graphs are shown to satisfy a “vertex-deletion strictly decreases isolation number” property. Consequently, $limsup_{n oinfty} f(n)/n = 1/4$, establishing a tight asymptotic theory for the $P_3$-isolation number.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

The $3$-path isolation number of a connected $n$-vertex graph $G$, denoted by $ι(G,P_3)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects each $3$-vertex path of $G$, meaning that no two edges of $G-N[D]$ intersect. Zhang and Wu proved that $ι(G,P_3) leq 2n/7$ unless $G$ is a $3$-path or a $3$-cycle or a $6$-cycle. The bound is attained by infinitely many graphs having induced $6$-cycles. Huang, Zhang and Jin proved that if $G$ has no $6$-cycles, or $G$ has no induced $5$-cycles and no induced $6$-cycles, then $ι(G, P_3) leq n/4$ unless $G$ is a $3$-path or a $3$-cycle or a $7$-cycle or an $11$-cycle. They asked if the bound still holds asymptotically for connected graphs having no induced $6$-cycles. More precisely, taking $f(n)$ to be the maximum value of $ι(G,P_3)$ over all connected $n$-vertex graphs $G$ having no induced $6$-cycles, their question is whether $limsup_{n oinfty}frac{f(n)}{n} = frac{1}{4}$. We verify this by proving that $f(n) = left lfloor (n+1)/4 ight floor$. The proof hinges on further proving that if $G$ is such a graph and $ι(G, P_3) = (n+1)/4$, then $ι(G-v, P_3) < ι(G, P_3)$ for each vertex $v$ of $G$. This new idea promises to be of further use. We also prove that if the maximum degree of such a graph $G$ is at least $5$, then $ι(G,P_3) leq n/4$.

Problem

• Determine the 3-path isolation number for graphs without induced 6-cycles.
• Verify if the asymptotic bound for isolation number holds as n approaches infinity.
• Explore the relationship between maximum degree and the 3-path isolation number.

Innovation

• Uses vertex subset for 3-path isolation
• Proves bound for graphs without 6-cycles
• Introduces new vertex removal technique