Paths and Cycles in d-Dimensional Tori with Faults
نویسندگان
چکیده
This paper is concerned with the paths and the cycles in d-dimensional tori with faulty vertices and/or edge. Let fv be the number of faulty vertices and fe be the number of faulty edges. It is shown that in any non-bipartite d-dimensional k1 × k2 × · · · kd torus with ki ≥ 3 for each 1 ≤ i ≤ d, (a) if fv + fe ≤ 2d − 3, there is a fault-free spanning path between any pair of non-faulty vertices, and (b) if fv + fe ≤ 2d− 2, there is a fault-free spanning cycle. It is also shown that in any bipartite d-dimensional k1 × k2 × · · · kd torus with ki ≥ 4 for each 1 ≤ i ≤ d, (a) if fv + fe ≤ 2d − 2, there is a fault-free path of length at least N − 2fv − 1 between any pair of non-faulty vertices which belong to the different partite sets, and there is a fault-free path of length at least N − 2fv − 2 between any pair of non-faulty vertices which belong to the same partite set, and (b) if fv + fe ≤ 2d− 1 and fe ≤ 2d− 2, there is a fault-free cycle of length at least N − 2fv, and if fv = 0 and fe = 2d − 1 and all the fault edges are not incident to a common vertex, there is a fault-free spanning cycle, and if fv = 0 and fe = 2d− 1 and all the fault edges are incident to a common vertex, there is a fault-free cycle of length N − 2 where N is the number of vertices.
منابع مشابه
Topological Compression Factors of 2-Dimensional TUC4C8(R) Lattices and Tori
We derived explicit formulae for the eccentric connectivity index and Wiener index of 2-dimensional square-octagonal TUC4C8(R) lattices with open and closed ends. New compression factors for both indices are also computed in the limit N-->∞.
متن کاملEmbedding Cartesian Products of Graphs into de Bruijn Graphs
Integration of concepts for the parallelization of image processing algorithms into parallel compiler technology. Abstract Given a Cartesian product G = G 1 : : : G m (m 2) of nontrivial connected graphs G i and the n{dimensional base B de Bruijn graph D = D B (n), it is investigated whether or not G is a spanning subgraph of D. Special attention is given to graphs G 1 : : : G m which are relev...
متن کاملPlanning Robot Motion in a 2-D Region with Unknown Obstacles
The purpose of this paper is to present several algorithms for planning the motion of a robot in a two-dimensional region having obstacles whose shapes and locations are unknown. The convergence and efficiency of the algorithms are discussed and upper bounds for the lengths of paths generated by the different algorithms are derived and compared.
متن کاملMulti-Dimensional Interval Routing Schemes
Interval Routing Scheme (k-IRS) is a compact routing scheme on general networks. It has been studied extensively and recently been implemented on the latest generation INMOS Transputer Router chip. In this paper we introduce an extension of the Interval Routing Scheme k-IRS to the multi-dimensional case hk; di-MIRS, where k is the number of intervals and d is the number of dimensions. Whereas k...
متن کاملCharacterization of signed paths and cycles admitting minus dominating function
If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.
متن کامل