  Leaf-Removal process has been widely researched and applied in many
mathematical and physical fields to help understand the complex systems, and a
lot of problems including the minimal vertex-cover are deeply related to this
process and the Leaf-Removal cores. In this paper, based on the structural
features of the Leaf-Removal cores, a method named Core Influence is proposed
to break the graphs into No-Leaf-Removal-Core ones, which takes advantages of
identifying some significant nodes by localized and greedy strategy. By
decomposing the minimal vertex-cover problem into the Leaf-Removal cores
breaking process and maximal matching of the remained graphs, it is proved that
any minimal vertex-covers of the whole graph can be located into these two
processes, of which the latter one is a P problem, and the best boundary is
achieved at the transition point. Compared with other node importance indices,
the Core Influence method could break down the Leaf-Removal cores much faster
and get the no-core graphs by removing fewer nodes from the graphs. Also, the
vertex-cover numbers resulted from this method are lower than existing node
importance measurements, and compared with the exact minimal vertex-cover
numbers, this method performs appropriate accuracy and stability at different
scales. This research provides a new localized greedy strategy to break the
hard Leaf-Removal Cores efficiently and heuristic methods could be constructed
to help understand some NP problems.
