A set of vertices S in a graph G is a global dominating set (GDS) of G if S is a dominating set for both G and its complement G. The minimum cardinality of a global dominating set of G is the global domination number of G. We explore the effects of graph modifications on the global domination number. In particular, we explore edge removal, edge addition, and vertex removal.

A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4.

Let G be a graph. For k ≥ d ≥ 1, a k/d -coloring of G is a coloring c of vertices of G with colors 0, 1, 2, . . ., k - 1, such that d ≤ | c(x) - c(y) | ≤ k - d, whenever xy is an edge of G. We say that the circular chromatic number of G, denoted χc(G), is equal to the smallest k/d where a k/d -coloring exists. In [6], Pan and Zhu have given a function μ(g) that gives an upper bound for the circular-chromatic number for every K4-minor-free graph Gg of odd girth at least g, g ≥ 3. In [7], they have shown that their upper bound in [6] can not be improved by constructing a sequence of graphs approaching μ(g) asymptotically. We prove that for every odd integer g = 2k + 1, there exists a graph Gg ∈ G/K4 of odd girth g such that χc(Gg) = μ(g) if and only if k is not divisible by 3. In other words, for any odd g, the question of attainability of μ(g) is answered for all g by our results. Furthermore, the proofs [6] and [7] are long and tedious. We give simpler proofs for both of their results.

The complementary prism of a graph G is obtained from a copy of G and its complement G̅ by adding a perfect matching between the corresponding vertices of G and G̅. For any graph G, a set D ⊆ V (G) is a double dominating set (DDS) if that set dominates every vertex of G twice. The double domination number, denoted γ×2(G), is the cardinality of a minimum double dominating set of G. We have proven results on graphs of small order, specific families and lower bounds on γ×2(GG̅).

Let H be a graph. G is a subgraph of H if V (G) ⊆ V (H) and E(G) ⊆ E(H). The subgraphs of H can be used to determine whether H is planar, a line graph, and to give information about the chromatic number. In a recent work by Beeler and Jamison [3], it was shown that it is difficult to obtain an automorphic decomposition of a triangle-free graph. As many of their examples involve circulant graphs, it is of particular interest to find triangle-free subgraphs within circulants. As a cycle with at least four vertices is a canonical example of a triangle-free subgraph, we concentrate our efforts on these. In this thesis, we will state necessary and sufficient conditions for the existence of edge induced and vertex induced cycles within circulants.

In the study of design theory, there are eight orientations of the complete graph on three vertices with a pendant edge, K3∪{e}. Two of these are the 3-circuit with a pendant arc and the other six are transitive triples with a pendant arc. Necessary and sufficient conditions are given for decompositions, packings and coverings of the complete digraph with each of the six transitive triples with a pendant arc.

For a graph G=(V,E), a nonempty subset S contained in V is called a defensive alliance if for each v in S, there are at least as many vertices from the closed neighborhood of v in S as in V-S. If there are strictly more vertices from the closed neighborhood of v in S as in V-S, then S is a strong defensive alliance. A (strong) defensive alliance is called global if it is also a dominating set of G. The alliance partition number (respectively, strong alliance partition number) is the maximum cardinality of a partition of V into defensive alliances (respectively, strong defensive alliances). The global (strong) alliance partition number is defined similarly. For each parameter we give both general bounds and exact values. Our major results include exact values for the alliance partition number of grid graphs and for the global alliance partition number of caterpillars.

A graph G(a, k, t) is called an alphabet overlap graph where a, k, and t are positive integers such that 0 ≤ t < k and the vertex set V of G is defined as, V = {v : v = (v1v2...vk); vi ∊ {1, 2, ..., a}, (1 ≤ i ≤ k)}. That is, each vertex, v, is a word of length k over an alphabet of size a. There exists an edge between two vertices u, v if and only if the last t letters in u equal the first t letters in v or the first t letters in u equal the last t letters in v. We determine the chromatic number of G(a, k, t) for all k ≥ 3, t = k − 2, and a = 2; except when k = 7, 8, 9, and 11.

After being introduced to graph theory and realizing how it can be utilized to solve real-world problems, the author decided to create modules of study on graph theory appropriate for middle school students. In this thesis, four modules were developed in the area of graph theory: an Introduction to Terms and Definitions, Graph Families, Graph Operations, and Graph Coloring. It is written as a guide for middle school teachers to prepare teaching units on graph theory.

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