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<p>The focus is on the following graph-theoretic question associated with the simulation ofcomplete binary trees by faulty hypercubes: if a certain number of nodes or links areremoved from an n-cube, will an (n-1)-tree still exists as a subgraph? While the generalproblem of determining whether a k-tree, k>n, still exists when an arbitrary number ofnodes/links are removed from the n-cube is found to be NP-complete, an upper bound isfound on how many nodes/links can be removed and an (n-1)-tree still be guaranteed toexist. In fact, as a corollary of this, it is found that if no more than n-3 nodes/links areremoved from an (n-1)-subcube of the n-cube, an (n-1)-tree is also guaranteed to exist.</p>
Index Termsfault tolerant embedding; complete binary trees; hypercubes; graph-theoretic question;simulation; k-tree; NP-complete; upper bound; computational complexity; fault tolerantcomputing; hypercube networks; trees (mathematics)
S.J. Lee, M.Y. Chan, "Fault-Tolerant Embedding of Complete Binary Trees in Hypercubes", IEEE Transactions on Parallel & Distributed Systems, vol. 4, no. , pp. 277-288, March 1993, doi:10.1109/71.210811
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