Taiwanese Journal of Mathematics

Nonseparating Independent Sets of Cartesian Product Graphs

Fayun Cao and Han Ren

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 2 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1554170479

Digital Object Identifier
doi:10.11650/tjm/190303

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

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

Citation

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


Export citation

References

  • B. Escoffier, L. Gourvès and J. Monnot, Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs, J. Discrete Algorithms 8 (2010), no. 1, 36–49.
  • H. Fernau and D. F. Manlove, Vertex and edge covers with clustering properties: complexity and algorithms, J. Discrete Algorithms 7 (2009), no. 2, 149–167.
  • M. R. Garey and D. S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. 32 (1977), no. 4, 826–834.
  • J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1987.
  • Y. Huang and Y. Liu, Maximum genus and maximum nonseparating independent set of a $3$-regular graph, Discrete Math. 176 (1997), no. 1-3, 149–158.
  • Y. Li, Z. Yang and W. Wang, Complexity and algorithms for the connected vertex cover problem in $4$-regular graphs, Appl. Math. Comput. 301 (2017), 107–114.
  • S. Long and H. Ren, The decycling number and maximum genus of cubic graphs, J. Graph Theory 88 (2018), no. 3, 375–384.
  • B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2001.
  • H. Moser, Exact algorithms for generalizations of vertex cover, Fakulätfür Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 2005 Mas-ters thesis.
  • D. A. Pike and Y. Zou, Decycling Cartesian products of two cycles, SIAM J. Discrete Math. 19 (2005), no. 3, 651–663.
  • S. Ueno, Y. Kajitani and S. Gotoh, On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three, Discrete Math. 72 (1988), no. 1-3, 355–360.
  • T. Wanatabe, S. Kajita and K. Onaga, Vertex covers and connected vertex covers in $3$-connected graphs, in: 1991., IEEE International Sympoisum on Circuits and Systems, (1991), 1017–1020.
  • N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory Ser. B 26 (1979), no. 2, 217–225.