Core Influence Mechanism on Vertex-Cover Problem through Leaf-Removal-Core Breaking

Abstract: 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.