Upper bounds on the $k$-isolation number

TL;DR

This work studies the $k$-isolation number $\iota_k(G)$, the smallest size of a vertex set whose closed neighborhood removal yields a graph of maximum degree at most $k-1$. It derives sharp, leaf-parameterized upper bounds for general connected graphs and for trees: $\iota_k(G)\le \frac{n-\ell}{2}$ for connected graphs with $n\ge 3$ that are not stars, and, for trees, $\iota(T)\le \frac{n+\ell}{4}$ and $\iota_k(T)\le \frac{n+\ell}{2k+1}$ for $k\ge 2$, with these bounds sometimes strictly better than the classical $\iota_k(T)\le \frac{n}{k+2}$. The results are shown to be tight, with complete extremal characterizations and explicit constructions (including corona-like graphs and specific path-based trees) for when equality holds. The proofs combine leaf-stripping arguments, Ore-type domination bounds, and careful structural constructions to elucidate how leaf structure governs isolation efficiency. Overall, the paper advances a precise understanding of how graph topology, via leaves, constrains $k$-isolation numbers and provides sharp, computable bounds across general graphs and trees.

Abstract

The isolation number of a graph $G$ (also called the vertex-edge domination number of $G$), denoted by $ι(G)$, is the size of a smallest subset $D$ of the vertex set $V(G)$ of $G$ such that $G-N[D]$ (the graph obtained by deleting the closed neighbourhood $N[D]$ of $D$ from $G$) has no edges. For $k \geq 1$, the $k$-isolation number of $G$ is the size of a smallest subset $D$ of $V(G)$ such that the maximum degree of $G-N[D]$ is at most $k-1$. Thus, $ι_1(G) = ι(G)$. Let $n$ and $\ell$ be the number of vertices and the number of leaves of $G$, respectively. We show that if $n \geq 3$ and $G$ is connected, then $ι_k(G) \leq \frac{n - \ell}{2}$. We also show that if $G$ is a tree $T$, then $ι(T) \leq \frac{n + \ell}{4}$ and $ι_k(T) \leq \frac{n + \ell}{2k+1}$ for $k \geq 2$. These bounds together improve the inequality $ι_k(T) \leq \frac{n}{k+2}$ of Caro and Hansberg except that their inequality is better if $k \geq 2$ and $\frac{k-1}{k+2}n < \ell < \frac{k}{k+2}n$. Each of the new bounds is attainable if it is an integer. For each of them, we characterize all the graphs that attain it.

Figures (6)

• Figure 1: A graph $G$: white vertices are in $L(G)$ and black vertices in $G-L(G)$. Circled vertices form a dominating set of $G-L(G)$ and, consequently, an isolating set of $G$.
• Figure 2: Graphs $G$ with $\iota(G)=\frac{n-\ell}{2}$.
• Figure 3: An isolating set of $T$ is formed by $u_2$ and an isolating set of $T'$.
• Figure 4: An isolating set of $T$ is formed by $u_2$ and an isolating set of $T'$.
• Figure 5: Trees belonging to the family $\mathcal{F}$ with $r$ copies of $P_3$ (left) and with $r$ copies of $P_3$ and $s\ge 1$ copies of $P_4$.

Theorems & Definitions (7)

• proof
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• proof : Proof of Theorem \ref{['thm:orderminusleaves_k']}
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