Upper bounds on the $k$-isolation number
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912231783923712 |
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| author | Borg, Peter Lemańska, Magdalena Mora, Mercè Souto-Salorio, María José |
| author_facet | Borg, Peter Lemańska, Magdalena Mora, Mercè Souto-Salorio, María José |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_14653 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper bounds on the $k$-isolation number Borg, Peter Lemańska, Magdalena Mora, Mercè Souto-Salorio, María José Combinatorics 05C69 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. |
| title | Upper bounds on the $k$-isolation number |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2408.14653 |