Computational Geometry is a branch of Computer Science that focuses on algorithms solving geometric problems. Computational Geometry has applications in various fields such as Modern Engineering and Mathematics, Computer Graphics, Robotics, VLSI Design, Computer-Aided Design, Molecular Modeling, Metallurgy, Manufacturing, Textile Design, Forestry, and Statistics. Graph theory is useful in defining problems and determining structural relationships. Hundreds of interesting computational problems are stated in graphs. In this study, the term of a circuit that more tightly covers the vertex set of the Euclidean complete graph has been defined, and an algorithm has been proposed to find this circuit. The complexity of the proposed algorithm is O(n). The proposed Boundary Point term in our presented work given in [5] allows for the expansion of the defined set of Boundary Vertices, enabling a more tightly construction of the Circuit Covering the Graph. The proposed algorithm can be applied to geometric design problems.