The problem is clearly NP-hard in general case (since its solution gives an answer to the NP-complete problem of determining whether a given graph has a Hamiltonian cycle). The associated decision problem of determining whether K edges can be added to a given graph to produce a Hamiltonian graph is NP-complete.
The problem may be solved in polynomial time for certain classes of graphs, including series-parallel graphs and their generalizations, which include outerplanar graphs, as well as for a line graph of a tree or a cactus graph.
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