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Nonseparating independent sets of Cartesian product graphs. (English) Zbl 1434.05106
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.
MSC:
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C76 Graph operations (line graphs, products, etc.)
05C05 Trees
05C40 Connectivity
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