Hierarchical Decomposition Mechanism by K\"{o}nig-Eg\'{e}rvary Layer-Subgraph with Application on Vertex-Cover

K\"{o}nig-Eg\'{e}rvary (KE) graph and theorem provides useful tools and deep understanding in the graph theory, which is an essential way to model complex systems. KE properties are strongly correlated with the maximum matching problem and minimum vertex cover problem, and have been widely researched and applied in many mathematical, physical and theoretical computer science problems. In this paper, based on the structural features of KE graphs and applications of maximum edge matching, the concept named KE-layer structure of general graphs is proposed to decompose the graphs into several layers. To achieve the hierarchical decomposition, an algorithm to verify the KE graph is given by the solution space expression of Vertex-Cover, and the relation between multi-level KE graphs and maximal matching is illustrated and proved. Furthermore, a framework to calculate the KE-layer number and approximate the minimal vertex-cover is proposed, with different strategies of switching nodes and counting energy. The phase transition phenomenon between different KE-layers are studied with the transition points located, the vertex cover numbers got by this strategy have comparable advantage against several other methods, and its efficiency outperforms the existing ones just before the transition point. Also, the proposed method performs stability and satisfying accuracy at different scales to approximate the exact minimum coverage. The KE-layer analysis provides a new viewpoint to understand the structural organizations of graphs better, and its formation mechanism can help reveal the intrinsic complexity and establish heuristic strategy for large-scale graphs/systems recognition.Comment: 12 pages, 4 figure