Summary: The game of Nim as played on graphs was introduced in [ibid. 304, No. 1–3, 387–399 (2003; Zbl 1041.91018)] and extended in [Theor. Comput. Sci. 304, No. 1–3, 401–419 (2003; Zbl 1041.91019)] by M. Fukuyama. His papers detail the calculation of Grundy numbers for graphs under specific circumstances. We extend these results and introduce the strategy for even cycles. This paper examines a more general class of graphs by restricting the edge weight to one. We provide structural conditions for which there exist a winning strategy. This yields the solution for the complete graph.