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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.09795 |
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Table of 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.