有向图的双控制数研究

Over the past forty years, the research on graph theory has already appeared the tendency of extremely activity with the computer science and network technology development quickly, and domination theory has become one of the fastest developing fields within graph theory. As an important research field in graph theory, domination theory has extensive application in related fields, such as computer science, communication networks, coding theory, operations search, and social sciences. Although domination theory of digraphs has the strong application background, it is still in its infancy. Thus, there are still many problems worth of exploring and mining. In particular, the study on twin domination number of digraph is at starting stage. In the project, we choose the domintion parameters of the Cartesian product of digraph, circulant digraph and a class of line digraph as the main emphasis of our research. Meanwhile by making full use of special structural properties of some classical network, more accurate research results about their twin domination number will be given. The research contents and results of the project will provide cogent reference data for application in Engineering, enrich the results of the domimation theory of graphs, and provide the necessary research experience and mode of thinking for further research on the upper and lower bounds and the optimization problem of the relevant parameters about twin dominating set of digraphs.

近四十年来，随着计算机科学和网络通讯技术的飞速发展，图论研究也呈现出异常活跃的趋势，而控制数理论是其中发展最快的领域之一。图的控制数理论作为图论的一个重要研究方向，在相关学科领域，例如计算机科学、通讯网络、编码理论、运筹学以及社会学等领域具有广泛的应用。而有向图的控制理论有较强的应用背景且起步较晚，因此，还有很多问题值得去研究探索和挖掘。特别地，有向图的双控制数的研究才刚刚起步。本项目选择有向图的笛卡尔积有向图，循环有向图以及一类线有向图的双控制参数作为我们的研究重点，同时我们将充分利用一些经典网络特殊的结构性质，对其双控制问题给出更加精确的研究结果。本项目的研究内容将会为工程应用提供有力的参考数据，同时丰富图的控制理论的成果，为我们进一步研究一般有向图的双控制集相关参数的上下界问题及优化问题等提供必要的研究经验和方法思路。

图的控制数理论作为图论的一个重要研究方向，在相关学科领域，例如计算机科学、通讯网络、编码理论、运筹学以及社会学等领域具有广泛的应用。而有向图的控制理论有较强的应用背景且起步较晚，因此，还有很多问题值得去研究探索和挖掘。特别地，有向图的双控制数的研究才刚刚起步。本项目主要研究了有向图的笛卡尔积有向图，一类线有向图的双控制数，研究了有向图的强积的控制数，对双控制集进行了限定，讨论了全双控制集，并计算某些特殊有向图类的全双控制数。本项目的研究内容丰富了图的控制理论的成果，为我们进一步研究一般有向图的双控制集相关参数的上下界问题及优化问题等提供必要的研究经验和方法思路。