Restrained and Other Domination Parameters in Complementary Prisms.

In this thesis, we will study several domination parameters of a family of graphs known as complementary prisms. We will first present the basic terminology and definitions necessary to understand the topic. Then, we will examine the known results addressing the domination number and the total domination number of complementary prisms. After this, we will present our main results, namely, results on the restrained domination number of complementary prisms. Subsequently results on the distance - k domination number, 2-step domination number and stratification of complementary prisms will be presented. Then, we will characterize when a complementary prism is Eulerian or bipartite, and we will obtain bounds on the chromatic number of a complementary prism. We will finish the thesis with a section on possible future problems

Locating-Domination in Complementary Prisms.

Let G = (V (G), E(G)) be a graph and G̅ be the complement of G. The complementary prism of G, denoted GG̅, is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. A set D ⊆ V (G) is a locating-dominating set of G if for every u ∈ V (G)D, its neighborhood N(u)⋂D is nonempty and distinct from N(v)⋂D for all v ∈ V (G)D where v ≠ u. The locating-domination number of G is the minimum cardinality of a locating-dominating set of G. In this thesis, we study the locating-domination number of complementary prisms. We determine the locating-domination number of GG̅ for specific graphs and characterize the complementary prisms with small locating-domination numbers. We also present bounds on the locating-domination numbers of complementary prisms

Italian Domination in Complementary Prisms

Let $G$ be any graph and let $\overline{G}$ be its complement. The complementary prism of $G$ is formed from the disjoint union of a graph $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. An Italian dominating function on a graph $G$ is a function such that $f \, : \, V \to \{ 0,1,2 \}$ and for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N(v)} f(u) \geq 2$. The weight of an Italian dominating function is the value $f(V)=\sum_{u \in V(G)}f(u)$. The minimum weight of all such functions on $G$ is called the Italian domination number. In this thesis we will study Italian domination in complementary prisms. First we will present an error found in one of the references. Then we will define the small values of the Italian domination in complementary prisms, find the value of the Italian domination number in specific families of graphs complementary prisms, and conclude with future problems

Domination in Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma(G)$ denote the domination number of $G$. It is readily seen that $\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.Comment: 18 pages, 8 figure