Matching Preclusion and Conditional Matching Preclusion Problems for Twisted Cubes

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find the matching preclusion number and the conditional matching preclusion number for twisted cubes, an improved version of the well-known hypercube. Moreover, we also classify all the optimal matching preclusion sets

On the extremal number of edges in hamiltonian connected graphs

AbstractAssume that n and δ are positive integers with 3≤δ<n. Let hc(n,δ) be the minimum number of edges required to guarantee an n-vertex graph G with minimum degree δ(G)≥δ to be hamiltonian connected. Any n-vertex graph G with δ(G)≥δ is hamiltonian connected if |E(G)|≥hc(n,δ). We prove that hc(n,δ)=C(n−δ+1,2)+δ2−δ+1 if δ≤⌊n+3×(nmod2)6⌋+1, hc(n,δ)=C(n−⌊n2⌋+1,2)+⌊n2⌋2−⌊n2⌋+1 if ⌊n+3×(nmod2)6⌋+1<δ≤⌊n2⌋, and hc(n,δ)=⌈nδ2⌉ if δ>⌊n2⌋

Fault-tolerant hamiltonian connectedness of cycle composition networks

Abstract It is important for a network to tolerate as many faults as possible. With the graph representation of an interconnection network, a k-regular hamiltonian and hamiltonian connected network is super fault-tolerant hamiltonian if it remains hamiltonian after removing up to k À 2 vertices and/or edges and remains hamiltonian connected after removing up to k À 3 vertices and/or edges. Super fault-tolerant hamiltonian networks have an optimal flavor with regard to the fault-tolerant hamiltonicity and fault-tolerant hamiltonian connectivity. For this reason, a cycle composition framework was proposed to construct a (k + 2)-regular super fault-tolerant hamiltonian network based on a collection of n k-regular super fault-tolerant hamiltonian networks containing the same number of vertices for n P 3 and k P 5. This paper is aimed to emphasize that the cycle composition framework can be still applied even when k = 4

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))