  We consider how to directly extract a road map (also known as a topological
representation) of an initially-unknown 2-dimensional environment via an online
procedure that robustly computes a retraction of its boundaries. In this
article, we first present the online construction of a topological map and the
implementation of a control law for guiding the robot to the nearest unexplored
area, first presented in [1]. The proposed method operates by allowing the
robot to localize itself on a partially constructed map, calculate a path to
unexplored parts of the environment (frontiers), compute a robust terminating
condition when the robot has fully explored the environment, and achieve loop
closure detection. The proposed algorithm results in smooth safe paths for the
robot's navigation needs. The presented approach is any time algorithm that has
the advantage that it allows for the active creation of topological maps from
laser scan data, as it is being acquired. We also propose a navigation strategy
based on a heuristic where the robot is directed towards nodes in the
topological map that open to empty space. We then extend the work in [1] by
presenting a topology matching algorithm that leverages the strengths of a
particular spectral correspondence method [2], to match the mapped environments
generated from our topology-making algorithm. Here, we concentrated on
implementing a system that could be used to match the topologies of the mapped
environment by using AOF Skeletons. In topology matching between two given maps
and their AOF skeletons, we first find correspondences between points on the
AOF skeletons of two different environments. We then align the (2D) points of
the environments themselves. We also compute a distance measure between two
given environments, based on their extracted AOF skeletons and their topology,
as the sum of the matching errors between corresponding points.
