Taiwanese Journal of Mathematics

Nonseparating Independent Sets of Cartesian Product Graphs

Fayun Cao and Han Ren

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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.

Article information

Taiwanese J. Math., Advance publication (2019), 17 pages.

First available in Project Euclid: 2 April 2019

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Primary: 05C05: Trees 05C69: Dominating sets, independent sets, cliques 05C70: Factorization, matching, partitioning, covering and packing

nonseparating independent set connected vertex cover hypercube Cartesian product of two cycles spanning tree Xuong-tree


Cao, Fayun; Ren, Han. Nonseparating Independent Sets of Cartesian Product Graphs. Taiwanese J. Math., advance publication, 2 April 2019. doi:10.11650/tjm/190303. https://projecteuclid.org/euclid.twjm/1554170479

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