Vertex covering problems of fuzzy graphs and their application in CCTV installation

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.