Graphs are common data structures which widely used for information storage and retrieval. Occasionally some vertices of a graph contain specific features or information, which we value in their effect. We consider modeling this effect formally, and we devise two super-fast algorithms to approximate the colored average degree. In the first method, we assume the information of each vertex is available; hence, the provided algorithm works with a 2+ϵ approximation factor. Eventually, we waive this assumption and find another algorithm with the same approximation factor, which computes the answer in the sublinear expected time.

Efficient distribution of service requests between fog and cloud nodes considering user mobility and fog nodes’ overload is an important issue of fog computing. This paper proposes a heuristic method for task placement considering the mobility of users, aiming to serve a higher number of requested services and minimize their response time. This method introduces a formula to overload prediction based on the entry-exit ratio of users and the estimated time required to perform current requests that are waiting in the queue of a fog node. Then, it provides a solution to avoid the predicted overloading of fog nodes by sending all delay-tolerant requests in the overloaded fog node’s queue to the cloud to reduce the time required for servicing delay-sensitive requests and to increase their acceptance rate. In addition, to prevent requests from being rejected when the mobile user leaves the coverage area of the current fog node, the requests in the current fog node’s queue will be transferred to the destination fog node. Simulation results indicate that the proposed method is effective in avoiding the overloading of the fog nodes and outperforms the existing methods in terms of response time and acceptance rate.

Let G = (V; E) be a graph. A double Roman dominating function (DRDF) on G is a function f: V! f0; 1; 2; 3g such that for every vertex v 2 V if f(v) = 0, then either there is a vertex u adjacent to v with f(u) = 3 or there are vertices x and y adjacent to v with f(x) = f(y) = 2 and if f(v) = 1, then there is a vertex u adjacent to v with f(u)  2. A DRDF f on G is a total DRDF (TDRDF) if for any v 2 V with f(v) > 0 there is a vertex u adjacent to v with f(u) > 0. The weight of f is the sum f(V ) = P v2V f(v). The minimum weight of a TDRDF on G is the total double Roman domination number of G. In this paper, we give a linear algorithm to compute the total double Roman domination number of a given tree.

The Internet of Things uses the cloud to process information received from electronic devices. Powerful servers located far from the sensors perform the processing. IoT devices send requests to the cloud and receive results from it. In some IoT applications, response time and latency are important. Therefore, latency should be reduced as much as possible. Sending information to the cloud itself entails latency. Therefore, the use of fog along with the cloud plays a fundamental role in the IoT. The use of fog and cloud computing in the field of IoT is a significant topic for researchers. To facilitate this process, software-defined networks have emerged as a vital component. These networks enable centralized control and management of the network. They also ensure optimal resource utilization and seamless connectivity by dynamically directing data flows to fog or cloud resources based on real-time conditions. Fog computing refers to the deployment of resources near the network sensor. By doing so, fog computing aims to reduce latency and bandwidth usage while improving overall system performance. It does this by utilizing local computing capabilities to process data. This research uses the proposed architecture for IoT networks and modeling different parts of this architecture using queuing theory to reduce response time using fog-cloud computing and obtain network quality of service parameters through mathematical analysis. In the following, the residual energy and latency comparison graphs for health and lighting applications as well as the use and non-use of fog-cloud computing are plotted. The graphs show that the use of fog-cloud computing reduces response time and the lighting application using fog has more residual energy and the health application has less latency. The simulation was performed using NS2 software in the smart home application.