<p>This paper gives classification results on the behavior of non-negative traveling waves, including their weak meaning, in the Fisher-KPP equation with power nonlinear degenerate diffusion in space one dimension. It gives information on the existence, shape and asymptotic form of the front-type traveling waves, unbounded and traveling waves with singularity. This is obtained by classifying all connecting orbits, including those to infinity, of the two-dimensional system of ordinary differential equations that characterize the traveling wave, using Poincaré compactification, one of the dynamical systems theory and geometric approaches used in this paper. Combined with the concept of weak solutions proposed in our previous studies and the results on changes in the qualitative characteristics of traveling waves due to nonlinear degenerate diffusion, the results indicate the influence of power nonlinear degenerate diffusion on the shape changes and classification results of traveling waves.</p>