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\(K_{1,2}\)-Isolation Number of Claw-Free Cubic Graphs

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Abstract

Let G be a graph and \({\mathcal {F}}\) be a family of connected graphs. A subset S of G is called an \({\mathcal {F}}\)-isolating set if \(G-N[S]\) contains no member in \({\mathcal {F}}\) as a subgraph, and the minimum cardinality of an \({\mathcal {F}}\)-isolating set of graph G is called the \({\mathcal {F}}\)-isolation number of graph G, denoted by \(\iota (G,{\mathcal {F}})\). For simplicity, let \(\iota (G,\{K_{1,k+1}\})=\iota _k(G)\). Thus, \(\iota _1(G)\) is the cardinality of a smallest set S such that \(G-N[S]\) consists of \(K_1\) and \(K_2\) only. In this paper, we prove that for any claw-free cubic graph G of order n, \(\iota _1(G)\le \frac{n}{4}\). The bound is sharp.

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Ürümqi, 830046, People’s Republic of China

    Yueqin Yin, Xinhui An & Baoyindureng Wu

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  1. Yueqin Yin
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  2. Xinhui An
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  3. Baoyindureng Wu
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Correspondence to Xinhui An.

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Communicated by Rosihan M. Ali.

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This work is supported by NSFC (Grant Nos. 11801487, 12061073, 11961067).

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Yin, Y., An, X. & Wu, B. \(K_{1,2}\)-Isolation Number of Claw-Free Cubic Graphs. Bull. Malays. Math. Sci. Soc. 47, 75 (2024). https://doi.org/10.1007/s40840-024-01672-w

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  • DOI: https://doi.org/10.1007/s40840-024-01672-w

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