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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References:

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