Solution to a problem on isolation of cliques in uniform hypergraphs

2025-12-31

AI Summary

This study investigates the isolation number of complete subhypergraphs $K_k^r$ (with $2 \leq r \leq k$) in $r$-uniform hypergraphs, defined as the minimum size of a vertex set whose closed neighborhood intersects every copy of $K_k^r$. By extending classical results from the graph case ($r=2$) and employing tools from extremal combinatorics, closed neighborhood analysis, and structural decomposition, the work fully characterizes the extremal structures for all parameter ranges and resolves an open problem posed by Li, Zhang, and Ye. The main result establishes that for any connected $r$-uniform hypergraph on $n$ vertices, the isolation number is at most $\frac{n}{k+1}$, with equality holding if and only if the hypergraph is itself a $K_k^r$, or when $k=r=2$ and the graph is a 5-cycle.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

A copy of a hypergraph $F$ is called an $F$-copy. Let $K_k^r$ denote the complete $r$-uniform hypergraph whose vertex set is $[k] = \{1, \dots, k\}$ (that is, the edges of $K_k^r$ are the $r$-element subsets of $[k]$). Given an $r$-uniform $n$-vertex hypergraph $H$, the $K_k^r$-isolation number of $H$, denoted by $\iota(H, K_k^r)$, is the size of a smallest subset $D$ of the vertex set of $H$ such that the closed neighbourhood $N[D]$ of $D$ intersects the vertex sets of the $K_k^r$-copies contained by $H$ (equivalently, $H-N[D]$ contains no $K_k^r$-copy). In this note, we show that if $2 \leq r \leq k$ and $H$ is connected, then $\iota(H, K_k^r) \leq \frac{n}{k+1}$ unless $H$ is a $K_k^r$-copy or $k = r = 2$ and $H$ is a $5$-cycle. This solves a recent problem of Li, Zhang and Ye. The result for $r = 2$ (that is, $H$ is a graph) was proved by Fenech, Kaemawichanurat and the author, and is used to prove the result for any $r$. The extremal structures for $r = 2$ were determined by various authors. We use this to determine the extremal structures for any $r$.

Problem

• isolation number
• uniform hypergraph
• complete hypergraph
• extremal structure

Innovation

• hypergraph isolation
• clique isolation number
• extremal structure
• uniform hypergraph
• K_k^r-copy