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Pose graph optimization is a special case of the simultaneous localization and mapping problem where the only variables to be estimated are pose variables and the only measurements are inter-pose constraints. The vast majority of pose graph optimization techniques are vertex based (variables are robot poses), but recent work has parameterized the pose graph optimization problem in a relative fashion (variables are the transformations between poses) that utilizes a minimum cycle basis to maximize the sparsity of the problem. We explore the construction of a cycle basis in an incremental manner while maximizing the sparsity. We validate an algorithm that constructs a sparse cycle basis incrementally and compare its performance with a minimum cycle basis. Additionally, we present an algorithm to approximate the minimum cycle basis of two graphs that are sparsely connected as is common in multi-agent scenarios. Lastly, the relative parameterization of pose graph optimization has been limited to using rigid body transforms on SE(2) or SE(3) as the constraints between poses. We introduce a methodology to allow for the use of lower-degree-of-freedom measurements in the relative pose graph optimization problem. We provide extensive validation of our algorithms on standard benchmarks, simulated datasets, and custom hardware.