For theoretical problems sure, you want to solve that problem, though a lot of solutions rely on analyzing a partial graph with some edges removed

For practical problems, changing the problem might be much cheaper/easier than keeping to the original one

> We take all the destinations that have an odd number of connections in this tree (Euler proved there must be an even number of these), and carefully pair them up. Because all the destinations now have an even number of edges, we’ve created an Eulerian graph, so we create a route that crosses each edge exactly once.