Beyond Rings: Gathering in 1-Interval Connected Graphs

We examine the problem of gathering [Formula: see text] agents (or multi-agent rendezvous) in dynamic graphs which may change in every round. We consider a variant of the [Formula: see text]-interval connectivity model [9] in which all instances (snapshots) are always connected spanning subgraphs of an underlying graph, not necessarily a clique. The agents are identical and not equipped with explicit communication capabilities, and are initially arbitrarily positioned on the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the underlying graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering, where the agents are allowed to gather either at the same node, or at two adjacent nodes. Our goal is to characterize the class of 1-interval connected graphs and initial configurations in which the problem is solvable, both with and without homebases. On the negative side we show that when the underlying graph contains a spanning bicyclic subgraph and satisfies an additional connectivity property, weak gathering is unsolvable, thus we concentrate mainly on unicyclic graphs. As we show, in most instances of initial agent configurations, the agents must meet on the cycle. This adds an additional difficulty to the problem, as they need to explore the graph and recognize the nodes that form the cycle. We provide a deterministic algorithm for the solvable cases of this problem that runs in [Formula: see text] number of rounds.

Connectivity, indecomposable, and weakly reversible in S-posets

Over the past four decades an extensive literature covered the properties of [Formula: see text]-acts. However, only few studies had generalized some known properties of [Formula: see text]-acts to the [Formula: see text]-posets. The reversible, and indecomposable properties in [Formula: see text]-posets have been addressed previously but connectivity has not been defined in [Formula: see text]-posets yet. Connectivity property was found to be related to those of reversibility and flatness in the category of [Formula: see text]-acts. The primary objective of this paper is to define connectivity in the category of [Formula: see text]-posets for both versions: ordered “poconnected” and unordered “connected”. Examples are presented to show the difference between the two versions. The relationship between connectivity with other properties such as reversibility, and indecomposability had also been investigated. We show that the poconnected in [Formula: see text]-posets is always indecomposable, but the inverse is not true. We also find that the weakly reversible is always connected and indecomposable. These relations among these properties in [Formula: see text]-posets are different from their corresponding relations in [Formula: see text]-acts.

Methods for Formation of Telecommunication Network States Sets for Different Measures of Connectivity

Reliability, survivability, and stability analysis tasks are typical not only for telecommunications, but also for systems whose components are subject to one or more types of failures, such as transport, power, mechanical systems, integrated circuits, and even software. The logical approach involves the decomposition of the system into a number of small functional elements, and within telecommunications networks they are usually separate network devices (switches, routers, terminals, etc.), as well as communication lines between them (copper-core, fiber-optic, coaxial cables, wireless media, and other transmission media). Functional relationships also define logical relationships between the failures of individual elements and the failure of the network as a whole. The assumption is also used that device failures are relatively less likely than communication line failures, which implies using the assumption of absolute stability (reliability, survivability) of these devices. Model of a telecommunication network in the form of the generalized model of Erdos–Renyi is presented. In the context of the stability of the telecommunications network, the analyzed property is understood as the connectivity of the network in one form or another. Based on the concept of stochastic connectivity of a network, as the correspondence of a random graph of the connectivity property between a given set of vertices, three connectivity measures are traditionally distinguished: two-pole, multi-pole, and all-pole. The procedures for forming an arbitrary structure of sets of paths and trees for networks are presented, as well as their generalization of multipolar trees. It is noted that multipolar trees are the most common concept of relatively simple chains and spanning trees. Solving such problems will allow us to proceed to calculating the probability of connectivity of graphs for various connectivity measures.

An Improved Size Invariant (n, n) Extended Visual Cryptography Scheme

In this paper, the authors have presented a (n, n) extended visual cryptography scheme where n numbers of meaningful shares furnish a visually secret message. Initially they have converted a grayscale image into binary image using dithering method. Afterwards, they have incorporated pixel's eight neighboring connectivity property of secret image during formation of meaningful shares. The scheme is able to generate the shares without extending its size. This approach has enhanced the visual quality of the recovered secret image from n numbers of shares. The scheme has been tested with some images and satisfactory results are achieved. The scheme has improved the contrast of the recovered secret image than a related (n, n) extended visual cryptography scheme.

{1,2,3}-Restricted Connectivity of $(n,k)$-Enhanced Hypercubes

The connectivity of a graph is a classic measure for fault tolerance of the network. Restricted connectivity measure is a crucial subject for a multiprocessor system’s ability to tolerate fault processors, and improves the connectivity measurement accuracy. Furthermore, if a network possesses a restricted connectivity property, it is more reliable with a lower vertex failure rate compared with other networks. The $\left (n,k\right )$-dimensional enhanced hypercube, denoted by $Q_{n,k}$, a variant of hypercube, which is a well-known interconnection network. In this paper, we analyze the fault tolerant properties for $\left (n,k\right )$-enhanced hypercube, and establish the $1$-restricted connectivity of $Q_{n,k} (n\ge k+1)$ and $\{2,3\}$-restricted connectivity of $(n,k)$-enhanced hypercube $Q_{n,k} (n=k+1)$. Furthermore, we propose the tight upper bound of $\{2,3\}$-restricted connectivity of $Q_{n,k} (n> k+1)$. Moreover, we show many figures to better illustrate the process of the proofs.

Kempe-Locking Configurations

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

Optimal Spectrum Utilization and Flow Controlling In Heterogeneous Network with Reconfigurable Devices

Fairness provisioning in heterogeneous networks is a prime issue for high-rate data flow, wherein the inter-connectivity property among different communication devices provides higher throughput. In Hetnet, optimal resource utilization is required for efficient resource usage. Proper resource allocation in such a network led to higher data flow performance for real-time applications. In view of optimal resource allocation, a resource utilization approach for a reconfigurable cognitive device with spectrum sensing capability is proposed in this paper. The allocation of the data flow rate at device level is proposed for optimization of network fairness in a heterogeneous network. A dynamic approach of rate-inference optimization is proposed to provide fairness in dynamic data traffic conditions. The simulation results validate the improvement in offered quality in comparison to multi-attribute optimization.

An Improved Size Invariant (n, n) Extended Visual Cryptography Scheme

In this paper, the authors have presented a (n, n) extended visual cryptography scheme where n numbers of meaningful shares furnish a visually secret message. Initially they have converted a grayscale image into binary image using dithering method. Afterwards, they have incorporated pixel's eight neighboring connectivity property of secret image during formation of meaningful shares. The scheme is able to generate the shares without extending its size. This approach has enhanced the visual quality of the recovered secret image from n numbers of shares. The scheme has been tested with some images and satisfactory results are achieved. The scheme has improved the contrast of the recovered secret image than a related (n, n) extended visual cryptography scheme.

Zero-One Law for Connectivity in Superposition of Random Key Graphs on Random Geometric Graphs

We study connectivity property in the superposition of random key graph on random geometric graph. For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here strengthen recent work by the Krishnan et al.