Folding on Topological Graphs and Their Fundamental Group

We introduce the effect of folding on the fundamental groups of a connected topological graph and its dual graph. We will deduce the limit of folding on the fundamental group of a topological graph and its duality. We obtain the relations between the induced folding and the induced retraction on the fundamental group. We apply retraction and folding operations to reduce the size of the graph while preserving its essential structural properties, which can assist us in developing mathematical software.