For the route inspection problem, T should be chosen as the set of all odd-degree vertices. By the assumptions of the problem, the whole graph is connected (otherwise no tour exists), and by the handshaking lemma it has an even number of odd vertices, so a T-join always exists. Doubling the edges of a T-join causes the given graph to become an Eulerian multigraph (a connected graph in which every vertex has even degree), from which it follows that it has an Euler tour, a tour that visits each edge of the multigraph exactly once. This tour will be an optimal solution to the route inspection problem.
For the route inspection problem, T should be chosen as the set of all odd-degree vertices. By the assumptions of the problem, the whole graph is connected (otherwise no tour exists), and by the handshaking lemma it has an even number of odd vertices, so a T-join always exists. Doubling the edges of a T-join causes the given graph to become an Eulerian multigraph (a connected graph in which every vertex has even degree), from which it follows that it has an Euler tour, a tour that visits each edge of the multigraph exactly once. This tour will be an optimal solution to the route inspection problem.