## Senior Projects Spring 2014

#### Title

Robots Switching Positions on Graphs

Spring 2014

Mathematics

Ethan Bloch

#### Abstract/Artist's Statement

Two robots travel simultaneously, but without passing each other on the same vertex or edge, from vertex to vertex on a simple, connected graph at a rate of one unit of time for every edge traveled. When possible, the two robots switch places so that their initial pair of positions, called the initial configuration, is reversed; the initial position of each robot is the final position of the other. If the two robots can switch places, there are two ways to determine the best way to travel: either the robots take the least amount of time to switch places by traveling in a time-minimal route, or the combined number of edges they travel is minimized, when the robots travel in a length-minimal route. Sometimes a time-minimal route for an initial configuration is also a length-minimal route, and the configuration is called unequivocal, and sometimes not. When every possible initial configuration on a graph is unequivocal, the graph is called graph unequivocal. We give a set of criteria that, when there are precisely one or two paths between the initial positions of the two robots, determines when a configuration is unequivocal. Further, there are a few types of graphs that we prove are always unequivocal.

On-Campus only