Origami, the art of paper folding, inspires structural modelling in science and engineering. This work formalizes origami-inspired transformations using graph theory by representing foldable structures as finite-induced subgraphs of infinite 2D rectangular and triangular grid graphs. We define two operations -Folding and Unfolding- that iteratively transform these graphs into truncated forms. We investigate the Hamiltonian and Eulerian properties of the resulting graphs. Triangular grid graphs preserve Hamiltonicity under both Type 1 and 2 folding operations (Theorem 1), but lose Eulericity under Type 2 (Theorem 2). In rectangular grid graphs, Hamiltonicity holds under both folding types for even-sized grids, while fails Eulericity (Theorems 11, 12). Special triangular grid graphs - hexagonal, rhomboidal, trapezoidal, starred and parrellelogram-shaped retain Hamiltonian properties (Theorems 2, 5, 6, 10, 11). However, hexagonal and starred graphs are non-Eulerian, while others exhibit semi-Eulerian behaviour. Formal proofs are provided using Grinberg’s Condition and degree based lemmas. This graph-theoretic framework offers a provable, discrete approach to origami-inspired systems, with applications in data compression, robotic path planning and neural grid modelling.