AbstractA total vertex irregularity strength of a graph G, tvs(G), is the minimum positive integer k such that there is a mapping f from the union of vertex and edge sets of G to {1, 2, · · ·, k} and the weights of all vertices are distinct. The weight of a vertex in G is the sum of its vertex label and the labels of all edges that incident to it. It is known that tvs(Kn) = 2.In this paper, we construct graphs with tvs equal to 2 by removing as much as possible edges from Kn, with and without maintaining the outer cycle Cn of Kn. To do so, we give two algorithms to construct the graphs, and show that the tvs of the resulting graph is equal to 2.