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The problem of simultaneous localization and mapping (SLAM) is addressed using a graphical method. The main contributions are a computational complexity that scales well with the size of the environment, the elimination of most of the linearization inaccuracies, and a more flexible and robust data association. We also present a detection criteria for closing loops. We show how multiple topological constraints can be imposed on the graphical solution by a process of coarse fitting followed by fine tuning. The coarse fitting is performed using an approximate system. This approximate system can be shown to possess all the local symmetries. Observations made during the SLAM process often contain symmetries, that is to say, directions of change to the state space that do not affect the observed quantities. It is important that these directions do not shift as we approximate the system by, for example, linearization. The approximate system is both linear and block diagonal. This makes it a very simple system to work with especially when imposing global topological constraints on the solution. These global constraints are nonlinear. We show how these constraints can be discovered automatically. We develop a method of testing multiple hypotheses for data matching using the graph. This method is derived from statistical theory and only requires simple counting of observations. The central insight is to examine the probability of not observing the same features on a return to a region. We present results with data from an outdoor scenario using a SICK laser scanner.