Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M=(V,E,A)M=(V,E,A) be a mixed graph with vertex set VV, edge set EE and arc set AA. A cycle cover of MM is a family C={C1,…,Ck}C={C1,…,Ck} of cycles of MM such that each edge/arc of MM belongs to at least one cycle in CC. The weight of CC is ∑i=1k|Ci|. The minimum cycle cover problem   is the following: given a strongly connected mixed graph MM without bridges, find a cycle cover of MM with weight as small as possible. The Chinese postman problem   is: given a strongly connected mixed graph MM, find a minimum length closed walk using all edges and arcs of MM. These problems are NP-hard. We show that they can be solved in polynomial time if MM has bounded tree-width.