The main objective of this paper is to reveal the evolving traversable wormhole solutions in the context of modified $f(\mathcal{R}, \mathcal{G})$ gravity, which causes the gravitational interaction. These results are derived by applying the Karmarkar condition which creates wormhole geometry that meets the necessary conditions and connects two asymptotically flat areas of spacetime. The proposed study's main goal is to construct the wormhole structures by splitting the $f(\mathcal{R,G})$ gravity model in two forms. Firstly, we split the model into Exponential like $f(\mathcal{R})$ gravity model and a power law $f(\mathcal{G})$ gravity model and secondly we consider the Starobinsky $f(\mathcal{R})$ gravity model along with $f(\mathcal{G})$ gravity power law model. Besides, we address the feasibility of shape functions and structural analysis of wormhole structures for specific models. These models are then confined to be compatible with current experimental evidence. Further, the energy conditions of the wormhole are geometrically probed and it is proven that they adhere to the null energy conditions $(NEC)$ in areas close to the throat. Moreover, the fascinating aspect of this study involves conducting an examination and comparison of evolving wormhole geometries in the vicinity of the throat in our chosen models, utilizing two- and three-dimensional graphical representations aided by 2D and 3D graphical depiction. We observe that our shape function acquired through the Karmarkar technique yields validated wormhole configurations with even less exotic matter correlating to the proper choice of $f(\mathcal{R}, \mathcal{G})$ gravity models and acceptable free parameter values. In summary, we can conclude that our findings meet all the criteria for the existence of wormhole, affirming the viability and consistency of our study.
