Bar-visibility graphs were introduced in the 1970s as a method to test printed circuit boards. Wismath and independently Tomassia and Tollis provided characterizations in the mid-1980s. Unit bar-visibility graphs are a type of bar-visibility graph in which all bars have the same unit length. This length restriction was motivated by circuits in which all components have near the same length.

This dissertation provides general results of unit bar-visibility graphs and then focuses on unit bar-visibility graphs having unit bar layouts with width less than two. Two characterizations of these graphs are provided. One characterization provides information on the unit bar layout of such a graph, and the other is tailored to finding a polynomial time algorithms to determine whether a graph has such a unit bar layout. Finally, Chapter 5 addresses induced subgraphs of grids, a challenging subclass of graphs that will guide the design of a general polynomial time algorithm for recognition of these graphs.