Graph Models for Backbone Sets and Limited Packings in Networks

Here, a graph-theoretic approach is applied to some problems in networks, for example in wireless sensor networks (WSNs) where some sensor nodes should be selected to behave as a backbone/dominating set to support routing communications in an efficient and fault-tolerant way. Four different types of multiple domination (k-, k-tuple, α‎- and α‎-rate domination) are considered and recent upper bounds for cardinality of these types of dominating sets are discussed. Randomized algorithms are presented for finding multiple dominating sets whose expected size satisfies the upper bounds. Limited packings in networks are studied, in particular the k-limited packing number. One possible application of limited packings is a secure facility location problem when there is a need to place as many resources as possible in a given network subject to some security constraints. The last section is devoted to two general frameworks for multiple domination: <r,s>-domination and parametric domination. Finally, different threshold functions for multiple domination are considered.

Introduction

This chapter gives a brief overview of selected applications of graph theory, many of which gave rise to the development of graph theory itself. A range of such applications extends from puzzles and games to serious scientific and real-life problems, thus illustrating the diversity of applications. The first section is devoted to the six earliest applications of graph theory. The next section introduces so-called scale-free networks, which include the web graph, social and biological networks. The last section describes a number of graph-theoretic algorithms, which can be used to tackle a number of interesting applications and problems of graph theory.

Graph Models for Optimization Problems in Road Networks

Here two applications of graph theory are considered. The first is devoted to pedestrian safety, and the focus is on pedestrian safety in urban areas with respect to pedestrian-vehicle crashes. In particular, an algorithm for automated construction of a graph model for pavement networks is discussed. Then, an algorithm for finding a user-optimal path in a given pavement network is presented. This algorithm is based on three criteria: path safety, distance, and path complexity. The second part of this chapter is devoted to optimizing the placement of charging stations for electric vehicles in road networks. The placement of charging stations in road networks is modelled as a multiple domination problem on reachability graphs. This model takes into account a threshold for the remaining battery charge and provides some minimal choice for a travel direction to recharge the battery. Experimental evaluation and simulations for the proposed facility location model are given for real road networks of the cities of Boston and Dublin.

Emergency Response: Navigable Networks and Optimal Routing in Hazardous Indoor Environments

The extreme importance of emergency response in complex buildings during natural and human-induced disasters has been widely acknowledged. This chapter studies algorithms for safest routes and balanced routes in buildings where an extreme event with many epicentres is occurring. In a balanced route, a trade-off between route length and hazard proximity is made. Another algorithm is proposed for finding the optimal indoor routes for search and rescue teams. This is based on a novel approach integrating the Analytic Hierarchy Process (AHP), the propagation of hazard and other techniques, and where three criteria are used: hazard proximity, distance/travel time, and route complexity. The important feature of the algorithm is its ability to generate an optimal route depending on the user’s needs. Finally, a novel automated construction of the Variable Density Network (VDN) for determining egress paths in dangerous environments is discussed.

Graphs in Molecular Epidemiology

Graphs and networks are used in molecular epidemiology to model the evolution of viruses and their spread during outbreaks and epidemics. They are instrumental at different stages of the computational pipelines. This includes the inference of transmission networks using viral sequences sampled from infected individuals, and studies of selection and accumulation of mutations in viral populations and their interactions with hosts' immune systems. This chapter describes some algorithmic and graph-theoretic problems associated with these stages to illustrate the relevance of the concepts of graph theory to molecular epidemiology of viral infections. The chapter will demonstrate how graph-theoretic methods combined with the machinery of differential equations, the Bayesian inference, and computational genomics uncover hidden biological and epidemiological patterns of virus evolution and transmission.

Traffic Networks: Wardrop Equilibrium and Braess’ Paradox

The well-known Braess’ paradox illustrates situations when adding a new link to a transport network might lead to an equilibrium state in which travel times of users will increase. Here, Braess’ paradox and the equilibrium state are analysed in the classical network configuration introduced by Braess in 1968. This network configuration is of fundamental significance because Valiant and Roughgarden showed in 2006 that ‘the “global” behaviour of an equilibrium flow in a large random network is similar to that in Braess’ original four-node example. Moreover, the probability of Braess’ paradox occurring in the classical network configuration will be studied, with particular emphasis on the Erlang distribution of parameters of the travel time function. This distribution is important in the context of traffic networks. However, other distributions will be analysed as well because Braess’ paradox can be observed in various applied contexts such as telecommunication networks and power transmission networks.