The generalized 3-edge-connectivity of lexicographic product graphs

The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $\lambda_3(G \circ H)$. Moreover, all bounds are sharp.Comment: 14 page

Steiner Distance in Product Networks

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.Comment: 29 pages, 4 figure

Design, Analysis and Computation in Wireless and Optical Networks

abstract: In the realm of network science, many topics can be abstracted as graph problems, such as routing, connectivity enhancement, resource/frequency allocation and so on. Though most of them are NP-hard to solve, heuristics as well as approximation algorithms are proposed to achieve reasonably good results. Accordingly, this dissertation studies graph related problems encountered in real applications. Two problems studied in this dissertation are derived from wireless network, two more problems studied are under scenarios of FIWI and optical network, one more problem is in Radio- Frequency Identification (RFID) domain and the last problem is inspired by satellite deployment. The objective of most of relay nodes placement problems, is to place the fewest number of relay nodes in the deployment area so that the network, formed by the sensors and the relay nodes, is connected. Under the fixed budget scenario, the expense involved in procuring the minimum number of relay nodes to make the network connected, may exceed the budget. In this dissertation, we study a family of problems whose goal is to design a network with “maximal connectedness” or “minimal disconnectedness”, subject to a fixed budget constraint. Apart from “connectivity”, we also study relay node problem in which degree constraint is considered. The balance of reducing the degree of the network while maximizing communication forms the basis of our d-degree minimum arrangement(d-MA) problem. In this dissertation, we look at several approaches to solving the generalized d-MA problem where we embed a graph onto a subgraph of a given degree. In recent years, considerable research has been conducted on optical and FIWI networks. Utilizing a recently proposed concept “candidate trees” in optical network, this dissertation studies counting problem on complete graphs. Closed form expressions are given for certain cases and a polynomial counting algorithm for general cases is also presented. Routing plays a major role in FiWi networks. Accordingly to a novel path length metric which emphasizes on “heaviest edge”, this dissertation proposes a polynomial algorithm on single path computation. NP-completeness proof as well as approximation algorithm are presented for multi-path routing. Radio-frequency identification (RFID) technology is extensively used at present for identification and tracking of a multitude of objects. In many configurations, simultaneous activation of two readers may cause a “reader collision” when tags are present in the intersection of the sensing ranges of both readers. This dissertation ad- dresses slotted time access for Readers and tries to provide a collision-free scheduling scheme while minimizing total reading time. Finally, this dissertation studies a monitoring problem on the surface of the earth for significant environmental, social/political and extreme events using satellites as sensors. It is assumed that the impact of a significant event spills into neighboring regions and there will be corresponding indicators. Careful deployment of sensors, utilizing “Identifying Codes”, can ensure that even though the number of deployed sensors is fewer than the number of regions, it may be possible to uniquely identify the region where the event has taken place.Dissertation/ThesisDoctoral Dissertation Computer Science 201