A set D of a graph G = (V,E) is a dominating set if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. A dominating set D is called a complementary tree dominating set if the induced sub graph < V −D > is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domin...

A graph X is called almost self-complementary with respect to a perfect matching I if it is isomorphic to the graph obtained from its complement Xc by removing the edges of I. A two-graph on a vertex set Ω is a collection T of 3-subsets of Ω such that each 4-subset of Ω contains an even number of elements of T . In this paper we investigate the relationship between self-complementary two-graphs...

We study square-complementary graphs, that is, graphs whose complement and square are isomorphic. We prove several necessary conditions for a graph to be square-complementary, describe ways of building new square-complementary graphs from existing ones, construct infinite families of square-complementary graphs, and characterize square-complementary graphs within various graph classes. The bipa...

A graph Γ is self-complementary if its complement is isomorphic to the graph itself. An isomorphism that maps Γ to its complement Γ is called a complementing isomorphism. The majority of this dissertation is intended to present my research results on the study of self-complementary vertex-transitive graphs. I will provide an introductory mini-course for the backgrounds, and then discuss four pr...

A graph X is called almost self-complementary with respect to a perfect matching I in the complement Xc if it is isomorphic to the graph obtained from Xc by removing the edges of I. In this paper we investigate the relationship between self-complementary two-graphs and double covers over complete graphs that are almost self-complementary with respect to their set of fibres. In particular, we cl...

The complementary prism of a graph Γ is the ΓΓ̄, which formed from union and its complement Γ̄ by adding an edge between each pair identical vertices in Γ̄. Vertex-transitive self-complementary graphs provide vertex-transitive prisms. It was recently proved author that ΓΓ̄ core, i.e. all endomorphisms are automorphisms, whenever vertex-transitive, self-complementary, either core or complete graph. ...