Proof of a conjecture on isolation of graphs with a universal vertex
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911000020647936 |
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| author | Borg, Peter Farrugia, Alastair |
| author_facet | Borg, Peter Farrugia, Alastair |
| contents | A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $ι(G,F)$, 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, $ι(G,K_1)$ is the domination number $γ(G)$ of $G$, and $ι(G,K_2)$ 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, $γ(F) = 1$ (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 $ι(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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture on isolation of graphs with a universal vertex Borg, Peter Farrugia, Alastair Combinatorics Discrete Mathematics 05C35, 05C69 A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $ι(G,F)$, 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, $ι(G,K_1)$ is the domination number $γ(G)$ of $G$, and $ι(G,K_2)$ 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, $γ(F) = 1$ (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 $ι(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. |
| title | Proof of a conjecture on isolation of graphs with a universal vertex |
| topic | Combinatorics Discrete Mathematics 05C35, 05C69 |
| url | https://arxiv.org/abs/2506.09290 |