Murty-Simon猜想及(全)控制临界图、边控制参数临界图性质的研究

图的控制集是反映图中重要节点的集合，有广泛应用背景，由此引出图（边）控制理论得到了广泛研究。同时边控制理论与图匹配、超图横贯等理论建立密切联系。近期，在网络容错性分析等研究有很好应用的（边）控制参数临界图研究非常活跃。本项目首先围绕Murty-Simon猜想，开展直径为2的3-全控制临界图刻画及边数下界的研究；其次，扩展到一般全控制、控制临界图，研究其直径、匹配相关及哈密尔顿等性质，推动两个关于直径猜想的解决；再者，提出部分边控制参数临界图概念，研究其在参数较小情况下的刻画；最后，开展一般边控制参数临界图结构性质研究，包括直径上界、哈密尔顿性等。这些挑战性、重要问题亟待解决。通过本项目提出新思想及有价值的研究方法，解决若干猜想和重要问题；首次提出部分边控制参数临界图并开展研究，奠定该方向理论研究基础；完善图的控制理论，特别是临界图方向，推动该方向发展。

A dominating set of a graph is a set of key vertices. It has a wide range of applications. From this, domination theory has been widely studied. At the same time, edge domination theory has established close relations with matchings, traversals of hypergraphs etc. Recently, research on (edge) domination-parameter critical graphs, which have been well applied in fault tolerance analysis of networks and others, is very active. Firstly, bound up with the Murty-Simon conjecture, the structural properties and characterization of 3-total-domination critical graphs of diameter 2 are studied; secondly, expanding to general (total-)domination critical graphs, we study their diameters, matchings and Hamilton properties, and promote the solutions to the two conjectures involving diameters; furthermore, this project proposes concepts of some edge domination critical graphs, and studies their characterizations in the case of small parameters; finally, it conducts research on the structural properties of general edge domination-parameter critical graphs, including upper bounds of their diameters, Hamiltonian, etc. These challenging problems and new problems need to be solved urgently. Through this project, new ideas and valuable research methods are proposed to solve some conjectures and important problems, partial edge domination-parameter critical graphs were firstly proposed and research was conducted to lay the foundation for the theory in this direction; studies on this project improve domination theory of graphs, especially on (edge) domination critical graphs, and push forward the development of this field.