On the Unique Tree Representation of Graphs
B. Jamison · ODU Digital Commons (Old Dominion University) · 2019
This dissertation investigates classes of graphs which admit tree representations unique up to isomorphism. The definitions of these classes are based on local properties of P4's, A template structure theorem is given which illustrates the nature of the local properties. The template theorem is instantiated for three different classes of graphs. Properties specific to each of the classes are used to produce a tree representation which can be used for graph computations, such as efficient resolution of graph isomorphism. Linear time algorithms for the recognition of three classes of graphs and for the construction of their tree representations are presented. The recognition algorithm also produces, as a by-product, a data structure which can be used for solving the four classic graph optimization problems in linear time.