A family of Hamiltonian and Hamiltonian connected graphs with fault tolerance

Abstract

Processor (vertex) faults and link (edge) faults may happen when a network is used, and it is meaningful to consider networks (graphs) with faulty processors and/or links. A k-regular Hamiltonian and Hamiltonian connected graph G is optimal fault-tolerant Hamiltonian and Hamiltonian connected if G remains Hamiltonian after removing at most k − 2 vertices and/or edges and remains Hamiltonian connected after removing at most k − 3 vertices and/or edges. In this paper, we investigate in constructing optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected graphs. Therefore, some of the generalized hypercubes, twistedcubes, crossed-cubes, and Möbius cubes are optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected.

Efficient Connectivity Testing of Hypercubic Networks with Faults

“…We believe that it would be interesting to find other classes of graphs for which the connectivity instance of Problem 1.1 has a positive solution. Natural candidates are other hypercubic networks [10] whose fault-tolerance has been investigated previously [2,11]. In some networks, transformations of walks are possible if we allow swaps on larger cycles (of bounded-size), e.g.…”

“…Next, considering the code of Procedure 6.2, the outer for-loop (line 1) is executed for every neighbor of f , while the inner for-loop (line 8) is executed for some vertices at distance two from f . Therefore, the total number of the inner loop executions is bounded by |F |n 2 Theorem 6.1. Given an integer n ≥ 1 and a set of vertices F of Q n , the problem whether the graph Q n − F is connected can be decided in O(n 3 |F |) time.…”