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February, 2020 Nonseparating Independent Sets of Cartesian Product Graphs
Fayun Cao, Han Ren
Taiwanese J. Math. 24(1): 1-17 (February, 2020). DOI: 10.11650/tjm/190303


A set of vertices $S$ of a connected graph $G$ is a nonseparating independent set if $S$ is independent and $G-S$ is connected. The nsis number $\mathcal{Z}(G)$ is the maximum cardinality of a nonseparating independent set of $G$. It is well known that computing the nsis number of graphs is NP-hard even when restricted to $4$-regular graphs. In this paper, we first present a new sufficient and necessary condition to describe the nsis number. Then, we completely solve the problem of counting the nsis number of hypercubes $Q_{n}$ and Cartesian product of two cycles $C_{m} \square C_{n}$, respectively. We show that $\mathcal{Z}(Q_{n}) = 2^{n-2}$ for $n \geq 2$, and $\mathcal{Z}(C_{m} \square C_{n}) = n + \lfloor (n+2)/4 \rfloor$ if $m = 4$, $m + \lfloor (m+2)/4 \rfloor$ if $n = 4$ and $\lfloor mn/3 \rfloor$ otherwise. Moreover, we find a maximum nonseparating independent set of $Q_{n}$ and $C_{m} \square C_{n}$, respectively.


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Fayun Cao. Han Ren. "Nonseparating Independent Sets of Cartesian Product Graphs." Taiwanese J. Math. 24 (1) 1 - 17, February, 2020.


Received: 16 November 2018; Revised: 19 February 2019; Accepted: 24 March 2019; Published: February, 2020
First available in Project Euclid: 2 April 2019

zbMATH: 07175536
MathSciNet: MR4053834
Digital Object Identifier: 10.11650/tjm/190303

Primary: 05C05, 05C69, 05C70

Rights: Copyright © 2020 The Mathematical Society of the Republic of China


Vol.24 • No. 1 • February, 2020
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