Given a digraph D = (V, A), with vertex set V and arc set A, a set S ⊆ V is a dominating set if for every vertex v in V \ S, there are a vertex u in S and an arc (u, v) from u to v. In this chapter we consider the counterparts in directed graphs of independent, dominating, independent dominating, and total dominating sets in undirected graphs.

A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The minimum cardinality of a paired dominating set of G is the paired domination number of G. This chapter presents a survey of the major results on paired domination with an emphasis on bounds for the paired domination number.

We study certain questions related to the performance of the Karp-Sipser algorithm on the sparse Erdös-Rényi random graph. The Karp-Sipser algorithm, introduced by Karp and Sipser [34] is a greedy algorithm which aims to obtain a near-maximum matching on a given graph. The algorithm evolves through a sequence of steps. In each step, it picks an edge according to a certain rule, adds it to the matching and removes it from the remaining graph. The algorithm stops when the remining graph is empty. In [34], the performance of the Karp-Sipser algorithm on the Erdös-Rényi random graphs G(n,M = [^cn/₂]) and G(n, p = ^c/_n), c > 0 is studied. It is proved there that the algorithm behaves near-optimally, in the sense that the difference between the size of a matching obtained by the algorithm and a maximum matching is at most o(n), with high probability as n → ∞. The main result of [34] is a law of large numbers for the size of a maximum matching in G(n,M = ^cn/₂) and G(n, p = ^c/_n), c > 0. Aronson, Frieze and Pittel [2] further refine these results. In particular, they prove that for c < e, the Karp-Sipser algorithm obtains a maximum matching, with high probability as n → ∞; for c > e, the difference between the size of a matching obtained by the algorithm and the size of a maximum matching of G(n,M = ^cn/₂) is of order Θ_{log n}(n^1/5), with high probability as n → ∞. They further conjecture a central limit theorem for the size of a maximum matching of G(n,M = ^cn/₂) and G(n, p = ^c/_n) for all c > 0. As noted in [2], the central limit theorem for c < 1 is a consequence of the result of Pittel [45]. In this thesis, we prove a central limit theorem for the size of a maximum matching of both G(n,M = ^cn/₂) and G(n, p = ^c/_n) for c > e. (We do not analyse the case 1 ≤ c ≤ e). Our approach is based on the further analysis of the Karp-Sipser algorithm. We use the results from [2] and refine them. For c > e, the difference between the size of a matching obtained by the algorithm and the size of a maximum matching is of order Θ_{log n}(n^1/5), with high probability as n → ∞, and the study [2] suggests that this difference is accumulated at the very end of the process. The question how the Karp-Sipser algorithm evolves in its final stages for c > e, motivated us to consider the following problem in this thesis. We study a model for the destruction of a random network by fire. Let us assume that we have a multigraph with minimum degree at least 2 with real-valued edge-lengths. We first choose a uniform random point from along the length and set it alight. The edges burn at speed 1. If the fire reaches a node of degree 2, it is passed on to the neighbouring edge. On the other hand, a node of degree at least 3 passes the fire either to all its neighbours or none, each with probability 1/2. If the fire extinguishes before the graph is burnt, we again pick a uniform point and set it alight. We study this model in the setting of a random multigraph with N nodes of degree 3 and α(N) nodes of degree 4, where α(N)/N → 0 as N → ∞. We assume the edges to have i.i.d. standard exponential lengths. We are interested in the asymptotic behaviour of the number of fires we must set alight in order to burn the whole graph, and the number of points which are burnt from two different directions. Depending on whether α(N) » √N or not, we prove that after the suitable rescaling these quantities converge jointly in distribution to either a pair of constants or to (complicated) functionals of Brownian motion. Our analysis supports the conjecture that the difference between the size of a matching obtained by the Karp-Sipser algorithm and the size of a maximum matching of the Erdös-Rényi random graph G(n,M = ^cn/₂) for c > e, rescaled by n^1/5, converges in distribution.