The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes

As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $\kappa(G, T)$ (resp. $T$-substructure connectivity $\kappa^{s}(G, T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. [6] showed $\kappa(Q_{n},K_{1,1})=\kappa^{s}(Q_{n},K_{1,1})=n-1$ and $\kappa(Q_{n},K_{1,r})=\kappa^{s}(Q_{n},K_{1,r})=\lceil\frac{n}{2}\rceil$ for $2\leq r\leq 3$ and $n\geq 3$. Sabir et al. [11] obtained that $\kappa(Q_{n},K_{1,4})=\kappa^{s}(Q_{n},K_{1,4})=\lceil\frac{n}{2}\rceil$ for $n\geq 6$, and for $n$-dimensional folded hypercube $FQ_{n}$, $\kappa(FQ_{n},K_{1,1})=\kappa^{s}(FQ_{n},K_{1,1})=n$, $\kappa(FQ_{n},K_{1,r})=\kappa^{s}(FQ_{n},K_{1,r})=\lceil\frac{n+1}{2}\rceil$ with $2\leq r\leq 3$ and $n\geq 7$. They proposed an open problem of determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general $r$. In this paper, we obtain that for each integer $r\geq 2$, $\kappa(Q_{n};K_{1,r})=\kappa^{s}(Q_{n};K_{1,r})=\lceil\frac{n}{2}\rceil$ and $\kappa(FQ_{n};K_{1,r})=\kappa^{s}(FQ_{n};K_{1,r})= \lceil\frac{n+1}{2}\rceil$ for all integers $n$ larger than $r$ in quare scale. For $4\leq r\leq 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases

A Local Diagnosis Algorithm for Hypercube-like Networks under the BGM Diagnosis Model

System diagnosis is process of identifying faulty nodes in a system. An efficient diagnosis is crucial for a multiprocessor system. The BGM diagnosis model is a modification of the PMC diagnosis model, which is a test-based diagnosis. In this paper, we present a specific structure and propose an algorithm for diagnosing a node in a system under the BGM model. We also give a polynomial-time algorithm that a node in a hypercube-like network can be diagnosed correctly in three test rounds under the BGM diagnosis model

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Mixed Dimensional Hierarchic Partitioned Analysis of Nonlinear Structural Systems

The use of most accurate 3D modelling in Finite Element analysis is too computationally expensive to be practical. The alternate simplification of 1D elements may sometimes compromise the accuracy of results. However, the detailed modelling of the critical parts and approximate modelling of the non-critical parts of the structure is often sufficient. A novel hierarchic domain partitioning approach has been developed in this study, which is being presented. The thesis begins with an introduction followed by a literature review. The domain partitioning approach is described next in which, parts of a structural system are removed and replaced by partition super elements. The removed parts are modelled separately with their partitioned boundary wrapped around by dual partition super elements. These partitions are analysed simultaneously using parallel computations. This domain partitioning approach increases the computational efficiency and allows the possibility of the use of differently dimensioned partitions. Using the domain partitioning approach, the complex non-linear structural systems can be subjected to static time-history, proportional loading, and dynamic loading. For dynamic analysis, eigenvalue analysis is required. The eigenvalue problem for large matrices is itself computationally expensive; however, a new parallel implementation of eigenvalue analysis is developed here using the partition super elements. In order to be able to use differently dimensioned partitions, a new dimensional coupling method has been developed. This is implemented with the help of a new master-slave element which has a single node as the master and all the nodes at the partitioned boundary as its slave nodes. This allows the possibility of using 3D brick elements inside a partition, whereas the other partitions at higher levels can use simplified 1D element models. The domain partitioning approach has been further enhanced by making it hierarchical, where, the partitions are further subdivided by replacing their parts with partition super elements and the removed parts modelled at further lower levels of partitioning. The hierarchical modelling is followed by a couple of case studies that demonstrate the applicability of the developed methods. The thesis ends with discussion and with the conclusion that the mixed dimensional hierarchic partitioning methods have greatly increased the computational efficiency of the finite element analysis. Some directions for the future research related to this work have been suggested

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Compilation of thesis abstracts, June 2007

NPS Class of June 2007This quarter’s Compilation of Abstracts summarizes cutting-edge, security-related research conducted by NPS students and presented as theses, dissertations, and capstone reports. Each expands knowledge in its field.http://archive.org/details/compilationofsis109452750