These are Eulerian and Hamiltonian circuits in the graph theory that have a wide-ranging role in diverse fields including logistics, network optimisation and computational biology. A circuit through the graph that uses every edge once is an Eulerian circuit; a circuit that uses every vertex once and returns to the starting point is a Hamiltonian circuit. The Hamiltonian circuit problem requires time intractable because Eulerian circuits convert to Hamiltonian circuits and vice versa are NP-complete recipes whereas Eulerian circuits are tractable (Garey & Johnson, 1979), still the task of transforming Eulerian circumvent to Hakan becomes complex but practically useful (Bondy & Murty, 2008).In this paper we explore the Eulerian-to-Hamiltonian circuit transformation, both in terms of the theoretical questions and computational realities of such a transformation. We also consider and solve a range of heuristic, optimization and exact algorithmic methods on different graph types: sparse, dense, and random graphs, as well as analyze their scalability, computational cost and quality of solution (Held & Karp, 1962; Diestel, 2017).Much of the attention is devoted to big graphs, in which graph topology (degree of vertices, sparsity, and connectivity) plays a decisive role in the performance of algorithms. The empirical results have shown that though the canonic approaches are appropriate to small graphs, the hybrid heuristic-optimization strategies seem to be an acceptable compromise on large examples. Real world implications are also described in the study, i.e. the study addresses applications in vehicle routing, network design and sequence assembly. Future larger scale experiments on scalable graph algorithm in dealing with dynamic, weighted, temporal networks on complex systems can be conducted on the basis of our findings.