Extremal Graph Theory is the particular subfield of Graph Theory concerned with maximizing and minimizing certain parameters associated with graphs and digraphs. Beginning with Mantel [46] and Ramsey [52], the study of extremal graphs was extended by Turán [56] into edge-maximum clique free graphs. Similar problems, particularly those in this thesis, are therefore referred to as Turán type problems.

Erdös, Hajnal, and Moon [21] generalized Turán's result even further, initiating the study of graph saturation. A graph H is G-free if it contains no subgraph isomorphic to G. H is G-saturated if it is edge-maximally G-free. Similarly, given a family [special characters omitted] of graphs, H is [special characters omitted]-saturated if it is G-free for every graph G in [special characters omitted] but the addition of any edge from the complement of H creates some graph in [special characters omitted]. Erdös, Hajnal, and Moon characterized, in particular, edge-minimum edge-maximally clique free graphs on a fixed number of vertices. Various others have characterized smallest edge-maximally G-free graphs for other classes of graphs G. In this thesis the concept of saturation is extended to oriented graphs, characterizations of extremal interval graphs and interval bigraphs are offered, minimal [special characters omitted]-saturated graphs are determined where [special characters omitted] is a family of subdivided graphs, a new saturation parameter on edge-colored graphs is introduced, and impartial and partizan games related to oriented graph saturation are examined.