K-core pruning process (or K-core decomposition, K-shell decomposition) is a well-known algorithm that has been used in thousands of papers by scientists from a broad range of research fields. Scientists use it to identify the most important nodes in a social network; find the most effective countries for crisis spreading in a global economic crisis; predict the structural collapse in mutualistic eco-systems; and locate the most influential spreaders in an epidemic process, etc. Despite the wide applications in dealing with the real-world problems, it also exhibits interesting critical behaviors that contain different kinds of phase transitions in the pruning process so that it is also appealing for many theoretical physicists. Due to the intrinsic mathematical complexity, yet there exist no clear theoretical results to the question of what the network is like during the pruning process. For the first time, we solve the mathematics and obtain the exact analytical results of the network for any given pruning step. With these exact analytical results, we can clearly depict even the finest details of the critical behavior in the process.