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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.09795 |
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| _version_ | 1866916651356651520 |
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| author | Boyer, Geoffrey Goddard, Wayne |
| author_facet | Boyer, Geoffrey Goddard, Wayne |
| contents | An isolating set of a graph is a set of vertices $S$ such that, if $S$ and its neighborhood is removed, only isolated vertices remain; and the isolation number is the minimum size of such a set. It is known that for every connected graph apart from $K_2$ and $C_5$, the isolation number is at most one-third the order and indeed such a graph has three disjoint isolating sets. In this paper we consider isolating sets where $S$ is required to be an independent set and call the minimum size thereof the independent isolation number. While for general graphs of order $n$ the independent isolation number can be arbitrarily close to $n/2$, we show that in bipartite graphs the vertex set can be partitioned into three disjoint independent isolating sets, whence the independent isolation number is at most $n/3$; while for $3$-colorable graphs the maximum value of the independent isolation number is $(n+1)/3$. We also provide a bound for $k$-colorable graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09795 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on Independent Isolation in Graphs Boyer, Geoffrey Goddard, Wayne Combinatorics An isolating set of a graph is a set of vertices $S$ such that, if $S$ and its neighborhood is removed, only isolated vertices remain; and the isolation number is the minimum size of such a set. It is known that for every connected graph apart from $K_2$ and $C_5$, the isolation number is at most one-third the order and indeed such a graph has three disjoint isolating sets. In this paper we consider isolating sets where $S$ is required to be an independent set and call the minimum size thereof the independent isolation number. While for general graphs of order $n$ the independent isolation number can be arbitrarily close to $n/2$, we show that in bipartite graphs the vertex set can be partitioned into three disjoint independent isolating sets, whence the independent isolation number is at most $n/3$; while for $3$-colorable graphs the maximum value of the independent isolation number is $(n+1)/3$. We also provide a bound for $k$-colorable graphs. |
| title | Bounds on Independent Isolation in Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.09795 |