A New Perspective on Vertex Connectivity

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $\lambda$ contains \floor{\lambda/2} edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoint spanning trees and that help us to better grasp the structure of graphs based on their vertex connectivity. The objective of the CDS partition problem is to partition the nodes of a graph into as many connected dominating sets as possible. The CDS packing problem is the corresponding fractional relaxation, where CDSs are allowed to overlap as long as this is compensated by assigning appropriate weights. CDS partition and CDS packing can be viewed as the counterparts of the well-studied edge-disjoint spanning trees, focusing on vertex disjointedness rather than edge disjointness. We constructively show that every $k$-vertex-connected graph with $n$ nodes has a CDS packing of size $\Omega(k/\log n)$ and a CDS partition of size $\Omega(k/\log^5 n)$. We prove that the $\Omega(k/\log n)$ CDS packing bound is existentially optimal. Using CDS packing, we show that if vertices of a $k$-vertex-connected graph are independently sampled with probability $p$, then the graph induced by the sampled vertices has vertex connectivity $\tilde{\Omega}(kp^2)$. Moreover, using our $\Omega(k/\log n)$ CDS packing, we get a store-and-forward broadcast algorithm with optimal throughput in the networking model where in each round, each node can send one bounded-size message to all its neighbors

A Survey on Energy-Efficient Strategies in Static Wireless Sensor Networks

A comprehensive analysis on the energy-efficient strategy in static Wireless Sensor Networks (WSNs) that are not equipped with any energy harvesting modules is conducted in this article. First, a novel generic mathematical definition of Energy Efficiency (EE) is proposed, which takes the acquisition rate of valid data, the total energy consumption, and the network lifetime of WSNs into consideration simultaneously. To the best of our knowledge, this is the first time that the EE of WSNs is mathematically defined. The energy consumption characteristics of each individual sensor node and the whole network are expounded at length. Accordingly, the concepts concerning EE, namely the Energy-Efficient Means, the Energy-Efficient Tier, and the Energy-Efficient Perspective, are proposed. Subsequently, the relevant energy-efficient strategies proposed from 2002 to 2019 are tracked and reviewed. Specifically, they respectively are classified into five categories: the Energy-Efficient Media Access Control protocol, the Mobile Node Assistance Scheme, the Energy-Efficient Clustering Scheme, the Energy-Efficient Routing Scheme, and the Compressive Sensing--based Scheme. A detailed elaboration on both of the basic principle and the evolution of them is made. Finally, further analysis on the categories is made and the related conclusion is drawn. To be specific, the interdependence among them, the relationships between each of them, and the Energy-Efficient Means, the Energy-Efficient Tier, and the Energy-Efficient Perspective are analyzed in detail. In addition, the specific applicable scenarios for each of them and the relevant statistical analysis are detailed. The proportion and the number of citations for each category are illustrated by the statistical chart. In addition, the existing opportunities and challenges facing WSNs in the context of the new computing paradigm and the feasible direction concerning EE in the future are pointed out