A linear algorithm for restructuring a graph

We propose an algorithm, linear in both running time and memory, that constructs a sequence of operations that transform any given directed graph with degree of any vertex at most two to any other given graph of the same type with minimal total cost. This sequence is called the shortest one. We allow four standard operations of re-gluing graphs with equal cost and two more additional operations of inserting and deleting a connected section of edges that also have equal cost. We prove that the algorithm finds a minimum with this restriction on the costs.