Characterizations of Certain Classes of Graphs and Matroids

Abstract

``If a theorem about graphs can be expressed in terms of edges and cycles only, it probably exemplifies a more general theorem about matroids." Most of my work draws inspiration from this assertion, made by Tutte in 1979. In 2004, Ehrenfeucht, Harju and Rozenberg proved that all graphs can be constructed from complete graphs via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. In Chapter 2, we consider the binary matroid analogue of each of these graph operations. We prove that the analogue of the result of Ehrenfeucht et. al. does not hold for binary matroids. However, we introduce a fourth operation that does enable the construction of all binary matroids from projective geometries. A graph in which every connected induced subgraph has a disconnected complement is called a cograph. Such graphs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. In Chapter 3, we define a 2-cograph to be a graph in which the complement of every 2-connected induced subgraph is not 2-connected. The class of 2-cographs is closed under induced minors. We characterize the class of non-2-cographs for which every proper induced minor is a 2-cograph. We further find the finitely many members of this class whose complements are also induced-minor-minimal non-2-cographs. Chapter 4 introduces binary comatroids, a matroid analogue of cographs. We identify all binary non-comatroids for which every proper flat is a binary comatroid. In addition, we extend our results to ternary matroids.