In graph theory, a vertex covering set \(V_\mathrm{C}\) is a set of vertices such that each edge of the graph is incident to at least one of the vertices of the set \(V_\mathrm{C}\). The problems related to vertex covering are called vertex covering problems. Many real-life problems contain a lot of uncertainties. To handle such uncertainties, concept of fuzzy set/graph is used. Here, we consider the covering problems of fuzzy graph to model some real-life problems. In this paper, a vertex covering problem is modeled as a series of linear and nonlinear programming problems with the help of basic graph-theoretic concept. In this model, the following objectives are considered: (1) the total number of facilities, the coverage area and total efficiency of all facilities are maximized, whereas (2) the total cost for the covering problem is minimized. Some new sets are defined and determined to make best decision on the basis of the features of facilities of the fuzzy system. An illustration is given to describe the whole model. Application of the said vertex covering problem to make a suitable decision for the placement of CCTVs in a city with the help of the developed formulations is given in a systematic way. To find the solutions, some algorithms are designed and the mathematical software ‘LINGO’ is used to keep the fuzziness of the parameters involved in the problems.

在图论中，覆盖集合\（V_ \ mathrm {C} \）的顶点是一组顶点，使得图的每个边均入射到集合\（V_ \ mathrm {C}的至少一个顶点上\）。与顶点覆盖有关的问题称为顶点覆盖问题。许多现实生活中的问题都包含很多不确定性。为了处理此类不确定性，使用了模糊集/图形的概念。在这里，我们考虑模糊图的覆盖问题来对一些实际问题进行建模。在本文中，借助基本的图论概念，将顶点覆盖问题建模为一系列线性和非线性规划问题。在此模型中，考虑了以下目标：（1）设施总数，覆盖面积和所有设施的总效率最大化，而（2）覆盖问题的总成本最小化。根据模糊系统的设施特征，定义和确定一些新集合以做出最佳决策。给出了描述整个模型的图示。借助于开发的公式，系统地给出了上述顶点覆盖问题的应用，以针对城市中闭路电视的放置做出合适的决定。为了找到解决方案，设计了一些算法，并使用数学软件“ LINGO”来保持问题所涉及参数的模糊性。