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.

A benzenoid is a molecule that can be represented as a graph. This graph is a fragment of the hexagon lattice. A dominating set $D$ in a graph $G$ is a set of vertices such that each vertex of the graph is either in $D$ or adjacent to a vertex in $D$. The domination number $\gamma=\gamma(G)$ of a graph $G$ is the size of a minimum dominating set. We will find formulas and bounds for the domination number of various special benzenoids, namely, linear chains $L(h)$, triangulenes $T_k$, and parallelogram benzenoids $B_{p,q}$. The domination ratio of a graph $G$ is $\frac{\gamma(G)}{n(G)}$, where $n(G)$ is the number of vertices of $G$. We will use the preceding results to prove that the domination ratio is no more than $\frac{1}{3}$ for the considered benzenoids. We conjecture that is true for all benzenoids.

A set S of vertices in a graph G=(V,E) is a dominating set if every vertex in V\S is adjacent to at least one vertex in S, and the minimum cardinality of a dominating set of G is the domination number of G. A vertex v in a dominating set S is said to be very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is very cost effective if every vertex in S is very cost effective. The minimum cardinality of a very cost effective dominating set of G is the very cost effective domination number of G. We first give necessary conditions for a graph to have equal domination and very cost effective domination numbers. Then we determine an upper bound on the very cost effective domination number for trees in terms of their domination number, and characterize the trees which attain this bound. lastly, we show that no such bound exists for graphs in general, even when restricted to bipartite graphs.

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. Our main results show that γR(GG) takes on a limited number of values in terms of the domination number of GG and the Roman domination numbers of G and G.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a double dominating set of G if for every v ∈ V(G)\S, v is adjacent to at least two vertices of S, and for every w ∈ S, w is adjacent to at least one vertex of S. The double domination number of G is the minimum cardinality of a double dominating set of G. We begin by determining the double domination number of complementary prisms of paths and cycles. Then we characterize the graphs G whose complementary prisms have small double domination numbers. Finally, we establish lower and upper bounds on the double domination number of GḠ and show that all values between these bounds are attainable.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a double dominating set of G if for every v ∈ V(G)\S, v is adjacent to at least two vertices of S, and for every w ∈ S, w is adjacent to at least one vertex of S. The double domination number of G is the minimum cardinality of a double dominating set of G. We begin by determining the double domination number of complementary prisms of paths and cycles. Then we characterize the graphs G whose complementary prisms have small double domination numbers. Finally, we establish lower and upper bounds on the double domination number of GḠ and show that all values between these bounds are attainable.

A set S of vertices in a graph G is a global dominating set of G if S simultaneously dominates both G and its complement Ḡ. The minimum cardinality of a global dominating set of G is the global domination number of G. We determine bounds on the global domination number of a graph and relationships between it and other domination related parameters.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a restrained dominating set of G if for every v € V(G) \S, v is adjacent to a vertex in S and a vertex in V(G) \S. The restrained domination number of G is the minimum cardinality of a restrained dominating set of G. We study restrained domination of complementary prisms. In particular, we establish lower and upper bounds on the restrained domination number of GḠ, show that the restrained domination number can be attained for all values between these bounds, and characterize the graphs which attain the lower bound.

A set S of vertices of a graph G = (V, E) is a dominating set if every vertex in V - S is adjacent to some vertex in 3. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. In June 2000, Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. However, a counterexample to this conjecture given in [6] suggests the modified conjecture that 1 ≤ sdγ(G) ≤ 4 for any graph G. It is also conjectured in [6] that for every graph G with minimum degree δ(G) ≥ 2, sdγ(G) ≤ δ(G) + 1. In this paper we extend several previous results and consider evidence in support of these two conjectures.