Efficient Generation of some Greedy Binary Gray codes

In 2013, Aaron Williams introduced the notion of a greedy Gray code algorithm and reinterpreted known Gray codes in a unified manner using greedy algorithms. Recently, this notion was further generalized and investigated by Merino, Mütze, and Williams in 2022, and by Merino and Mütze in 2024, in the context of generating the bases of a matroid or the spanning trees of a graph, among other combintorial structures. In this article, we investigate the existence of homogeneous greedy Gray codes for Fibonacci words and generalized Dyck prefixes. We also establish useful properties and provide efficient generation algorithms for them.