Label-guided graph exploration by a finite automaton

A finite automaton, simply referred to as a robot, has to explore a graph, that is, visit all the nodes of the graph. The robot has no a priori knowledge of the topology of the graph, nor of its size. It is known that for any k-state robot, there exists a graph of maximum degree 3 that the robot cannot explore. This article considers the effects of allowing the system designer to add short labels to the graph nodes in a preprocessing stage, for helping the exploration by the robot. We describe an exploration algorithm that, given appropriate 2-bit labels (in fact, only 3-valued labels), allows a robot to explore all graphs. Furthermore, we describe a suitable labeling algorithm for generating the required labels in linear time. We also show how to modify our labeling scheme so that a robot can explore all graphs of bounded degree, given appropriate 1-bit labels. In other words, although there is no robot able to explore all graphs of maximum degree 3, there is a robot R, and a way to color in black or white the nodes of any bounded-degree graph G, so that R can explore the colored graph G. Finally, we give impossibility results regarding graph exploration by a robot with no internal memory (i.e., a single-state automaton).