Graphical representation and hierarchical decomposition mechanism for vertex-cover solution space

NP problems act essential roles in modelling and analysing various complex systems, and representation learning of system individuals and relations has faced the kernel difficulty in understanding the complexity and solving the NP problems. In this paper, solution space organisation of minimum vertex-cover problem is deeply investigated using the famous König-Egérvary (KE) graph and theorem, in which a hierarchical decomposition mechanism named KE-layer structure of general graphs is proposed to reveal the complexity of vertex-cover. To achieve the graphical representation and hierarchical decomposition, an algorithm to verify the KE graph is given by the solution space expression of vertex-cover, and the relation between multi-layer 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 provided, with different strategies of switching nodes and counting energy. The phase transition phenomenon between different KE-layers is studied with the transition points located, and searching of vertex-cover got by this strategy presents comparable advantage against several other methods. Its efficiency outperforms the existing ones just before the transition point. The graphical representation and hierarchical decomposition provide a new perspective to illustrate the structural organisations of graphs better, and its formation mechanism can help reveal the intrinsic complexity and establish heuristic strategy for large-scale graphs/systems recognition.