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Panconnectivity and pancyclicity of hypercube-like interconnection networks with faulty elements. (English) Zbl 1115.68116

Summary: We deal with the graph \(G_0 \oplus G_1\) obtained from merging two graphs \(G_0\) and \(G_1\) with n vertices each by \(n\) pairwise nonadjacent edges joining vertices in \(G_0\) and vertices in \(G_1\). The main problems studied are how fault-panconnectivity and fault-pancyclicity of \(G_0\) and \(G_1\) are translated into fault-panconnectivity and fault-pancyclicity of \(G_0 \oplus G_1\), respectively. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of \(G_0 \oplus G_1\) connecting two lower dimensional networks \(G_0\) and \(G_1\). Applying our results to a class of hypercube-like interconnection networks called restricted HL-graphs, we show that in a restricted HL-graph \(G\) of degree \(m (\geq 3)\), each pair of vertices are joined by a path in \(G\backslash F\) of every length from \(2m-3\) to \(|V(G\backslash F)|-1\) for any set \(F\) of faulty elements (vertices and/or edges) with \(|F|\leq m-3\), and there exists a cycle of every length from 4 to \(|V(G\backslash F)|\) for any fault set \(F\) with \(|F|\leq m-2\).

MSC:

68R10 Graph theory (including graph drawing) in computer science
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