an:07175536
Zbl 1434.05106
Cao, Fayun; Ren, Han
Nonseparating independent sets of Cartesian product graphs
EN
Taiwanese J. Math. 24, No. 1, 1-17 (2020).
00447215
2020
j
05C69 05C70 05C76 05C05 05C40
nonseparating independent set; connected vertex cover; hypercube; Cartesian product of two cycles; spanning tree; Xuong-tree
Summary: 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.