A subset S of vertices in a graph G=(V,E) is a connected dominating set of G if every vertex of V\-S is adjacent to a vertex in S and the subgraph induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number γc(G). The girth g(G) is the length of a shortest cycle in G. We show that if G is a connected graph that contains at least one cycle, then γc(G)≥g(G)-2, and we characterize the graphs obtaining equality in this bound. We also establish various upper bounds on the connected domination number of a graph, as well as Nordhaus-Gaddum type results.

A subset S of vertices in a graph G=(V,E) is a connected dominating set of G if every vertex of V\-S is adjacent to a vertex in S and the subgraph induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number γc(G). The girth g(G) is the length of a shortest cycle in G. We show that if G is a connected graph that contains at least one cycle, then γc(G)≥g(G)-2, and we characterize the graphs obtaining equality in this bound. We also establish various upper bounds on the connected domination number of a graph, as well as Nordhaus-Gaddum type results.

A set S of vertices in a graph G is a connected dominating set of G if S dominates G and the subgraph induced by S is connected. We study the graphs for which adding any edge does not change the connected domination number.

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination numberγt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑u ∈ V (G) f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only ifγt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.

Let G be a graph with vertex set V and no isolated vertices. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. We present a method of building trees having a unique minimum semipaired dominating set.

Let G be a graph with vertex set V and no isolated vertices. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number γ pr2 (G) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph G of order n ≥ 3, then γ pr2 (G) ≤ 32 n, and we characterize the extremal graphs achieving equality in the bound.

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to at least one vertex in S. In this chapter, we present logical models called frameworks, each of which gives a different perspective of dominating sets.

Let G be a graph with vertex set V. A subset S ? V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.

A set D of vertices in a graph G is a 2-dominating set of G if every vertex in V − D has at least two neighbors in D. The 2-domination number of a graph G, denoted by γ2(G), is the minimum cardinality of a 2- dominating set of G. In this thesis, we discuss the 2-domination number of a special family of trees, called caterpillars. A caterpillar is a graph denoted by Pk(x1, x2, ..., xk), where xi is the number of leaves attached to the ith vertex of the path Pk. First, we present the 2-domination number of some classes of caterpillars. Second, we consider several types of complete caterpillars. Finally, we consider classification of caterpillars with respect to their spine length and 2-domination number.

For a graph G = (V (G), E (G)), a set S ~ V (G) dominates G if each vertex in V (G) \S is adjacent to a vertex in S. The domination number I (G) (independent domination number i (G)) of G is the minimum cardinality amongst its dominating sets (independent dominating sets). G is k-edge-domination-critical, abbreviated k-1- critical, if the domination number k decreases whenever an edge is added. Further, G is hamiltonian if it has a cycle that passes through each of its vertices. This dissertation assimilates research generated by two conjectures: Conjecture I. Every 3-1-critical graph with minimum degree at least two is hamiltonian. Conjecture 2. If G is k-1-critical, then I ( G) = i ( G). The recent proof of Conjecture I is consolidated and presented accessibly. Conjecture 2 remains open for k = 3 and has been disproved for k :::>: 4. The progress is detailed and proofs of new results are presented. / Mathematical Science / M. Sc. (Mathematics)