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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:
68R10Graph theory in connection with computer science (including graph drawing)
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