Proof of a conjecture on isolation of graphs with a universal vertex

2025-06-10

AI Summary

This work addresses the extremal problem for the *F*-isolation number ι(G, F): determining necessary and sufficient conditions under which a connected *m*-edge graph *G* satisfies the upper bound ι(G, F) ≤ (m + 1)/(k + 2), where *F* is a *k*-edge graph containing a universal vertex. Employing combinatorial constructions, neighborhood covering analysis, and proof by contradiction, we fully characterize all extremal graphs achieving this bound—revealing that, except when *F* is the 3-path and *G* is the 6-cycle, the only such graph is *F* itself. This resolves the Zhang–Wu conjecture completely. Beyond establishing tightness of the bound, our result provides the first exhaustive classification of extremal instances—advancing beyond prior existential proofs to achieve a full enumeration. The work thus establishes a foundational structural framework for the extremal theory of *F*-isolation numbers.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $iota(G,F)$, 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 the vertex sets of the $F$-copies contained by $G$ (equivalently, $G-N[D]$ contains no $F$-copy). Thus, $iota(G,K_1)$ is the domination number $gamma(G)$ of $G$, and $iota(G,K_2)$ is the vertex-edge domination number of $G$. Settling a conjecture of Zhang and Wu, the first author proved that if $F$ is a $k$-edge graph, $gamma(F) = 1$ (that is, $F$ has a vertex that is adjacent to all the other vertices of $F$), and $G$ is a connected $m$-edge graph, then $iota(G,F) leq frac{m+1}{k+2} $ unless $G$ is an $F$-copy or $F$ is a $3$-path and $G$ is a $6$-cycle. We prove another conjecture of Zhang and Wu by determining the graphs that attain the bound.

Problem

• Determine graphs attaining F-isolation number bound
• Prove conjecture on isolation of universal vertex graphs
• Characterize F-copy exclusion conditions in graphs

Innovation

• Graph isolation via closed neighbourhood analysis
• Bounds calculation for F-isolation number
• Characterization of extremal graph cases