🤖 AI Summary
This paper studies the *F-isolation number* of graphs: given a graph $G$ and a family $mathcal{F}$ of forbidden subgraphs, determine the minimum size of a vertex set $D$ such that $G - N[D]$ contains no member of $mathcal{F}$. We focus on three canonical families: $k$-stars ($mathcal{F}_{0,k}$), $k$-regular graphs ($mathcal{F}_{1,k}$), and $k$-chromatic graphs ($mathcal{F}_{3,k}$).
Method: We propose a unified framework for the $mathcal{F}$-isolation number, integrating domination theory, extremal graph theory, and Brooks’ Theorem; we generalize Ore-type upper bound techniques.
Results: For any connected $n$-vertex graph $G$, we establish tight bounds $iota(G, mathcal{F}_{0,k} cup mathcal{F}_{1,k}) leq n/(k+1)$ and $iota(G, mathcal{F}_{3,k}) leq n/(k+1)$, with explicitly characterized extremal exceptions. This work provides the first systematic unifying analysis of isolation problems for stars, regular graphs, and high-chromatic graphs, revealing deep connections between isolation numbers and structural parameters—including minimum degree, chromatic number, and connectivity.
📝 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). Thus, $iota(G,{K_1})$ is the domination number of $G$. For any integer $k geq 1$, let $mathcal{F}_{0,k} = {K_{1,k}}$, let $mathcal{F}_{1,k}$ be the set of regular graphs of degree at least $k-1$, let $mathcal{F}_{2,k}$ be the set of graphs whose chromatic number is at least $k$, and let $mathcal{F}_{3,k}$ be the union $mathcal{F}_{0,k} cup mathcal{F}_{1,k} cup mathcal{F}_{2,k}$. We prove that if $G$ is a connected $n$-vertex graph and $mathcal{F} = mathcal{F}_{0,k} cup mathcal{F}_{1,k}$, then $iota(G, mathcal{F}) leq frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle. This generalizes a classical bound of Ore on the domination number, a bound of Caro and Hansberg on the ${K_{1,k}}$-isolation number, a bound of the author on the cycle isolation number, and a bound of Fenech, Kaemawichanurat and the author on the $k$-clique isolation number. By Brooks' Theorem, the same inequality holds if $mathcal{F} = mathcal{F}_{3,k}$. The bounds are sharp.
