Graph-based Simultaneous Localization and Mapping (Graph-SLAM) formulates spatial mapping as a non-linear graph optimization problem across state space. Robot spatial poses form graph vertices linked by odometric motion constraints, while sensor observations link pose nodes to stationary landmark vertices. Detecting loop closures introduces spatial constraints between distant trajectory segments; non-linear least squares optimization (e.g. Levenberg-Marquardt) relaxes graph deformations to minimize cumulative error.