Let G= (V, E) be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function f: V∪ E→ { 0 , 1 , 2 } satisfying the condition every element x∈ V∪ E for which f(x) = 0 is adjacent or incident to at least one element y∈ V∪ E for which f(y) = 2. The weight of a MRDF f is ω(f) = ∑ x∈V∪Ef(x). The mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.

Degree constraints on the vertices of a path allow for the definitions of uphill and downhill paths. Specifically, we say that a path P = vi, v2,⋯ vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(v1+1). Conversely, a path π = u1, u2,⋯ uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(ui) ≤ deg(ui+1). The downhill domination number of a graph G is the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We give a polynomial time algorithm to find a minimum downhill dominating set and a minimum uphill dominating set for any graph.

Placing degree constraints on the vertices of a path yields the definitions of uphill and downhill paths. Specifically, we say that a path π = v1, v2, ⋯ vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(v1) ≥ deg(vi+1). Conversely, a path π = u1, u2, ⋯ uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(u1) ≤ deg(ui+1). The downhill domination number of a graph G is defined to be the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We explore the properties of these invariants and their relationships with other invariants. We also determine a Vizing-like result for the downhill (respectively, uphill) domination numbers of Cartesian products.

The domination subdivision number of a graph is the minimum number of edges that must be subdivided in order to increase the domination number of the graph. Likewise, the total domination subdivision number is the minimum number of edges that must be subdivided in order to increase the total domination number. First, this thesis provides a complete survey of established bounds on the domination subdivision number and the total domination subdivision number. Then in Chapter 4, new results regarding bounds on the total domination subdivision number are given. Finally, a characterization of the total domination subdivision number of caterpillars is presented in Chapter 5.

In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979-991]. We study the paired-domination version of the domination game which adds a matching dimension to the game. This game is played on a graph G by two players, named Dominator and Pairer. They alternately take turns choosing vertices of G such that each vertex chosen by Dominator dominates at least one vertex not dominated by the vertices previously chosen, while each vertex chosen by Pairer is a vertex not previously chosen that is a neighbor of the vertex played by Dominator on his previous move. This process eventually produces a paired-dominating set of vertices of G; that is, a dominating set in G that induces a subgraph that contains a perfect matching. Dominator wishes to minimize the number of vertices chosen, while Pairer wishes to maximize it. The game paired-domination number γgpr(G) of G is the number of vertices chosen when Dominator starts the game and both players play optimally. Let G be a graph on n vertices with minimum degree at least 2. We show that γgpr(G) ≤ 45 n, and this bound is tight. Further we show that if G is (C4, C5)-free, then γgpr(G) ≤ 43 n, where a graph is (C4, C5)-free if it has no induced 4-cycle or 5-cycle. If G is 2-connected and bipartite or if G is 2-connected and the sum of every two adjacent vertices in G is at least 5, then we show that γgpr(G) ≤ 34 n.

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n<2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n<2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

Let uxv be an induced path with center x in a graph G. The edge lifting of uv off x is defined as the action of removing edges ux and vx from the edge set of G, while adding the edge uv to the edge set of G. We study trees for which every possible edge lift changes the domination number. We show that there are no trees for which every possible edge lift decreases the domination number. Trees for which every possible edge lift increases the domination number are characterized.

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number yγ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt (G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1 ≤sdγt (T)≤3. In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.

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 γpr 2(G) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph of order n with minimum degree at least 2, then γpr2(G)≤12(n+1). Further, we show that if n≢3(mod4), then γpr2(G)≤12n, and we show that for every value of n≡3(mod4), there exists a connected graph G of order n with minimum degree at least 2 satisfying γpr2(G)=12(n+1). As a consequence of this result, we prove that every graph G of order n with minimum degree at least 3 satisfies γpr2(G)≤12n.