We consider the robot exploration of a planar graph-like world. The
robot's goal is to build a complete map of its environment. The
environment is modeled as an arbitrary undirected planar graph which
is initially unknown to the robot. The robot cannot distinguish
vertices and edges that it has explored from the unexplored ones. The
robot is assumed to be able to autonomously traverse graph edges,
recognize when it has reached a vertex, and enumerate edges incident
upon the current vertex. The robot cannot measure distances nor does
it have a compass, but it is equipped with a single marker that it can
leave at a vertex and sense if the marker is present at a newly
visited vertex. The total number of edges traversed while constructing
a map of a graph is used as a measure of performance. We present an
efficient algorithm for learning an unknown, undirected planar graph
by a robot equipped with one marker. One of the main results of this
paper is to show that our strategy leads to performance that is
typical linear in the size of the graph. Experimental results obtained
by running a large collection of example worlds are also presented.