TITLE: How to efficiently find a Hamiltonian cycle in a graph whose closure is complete?
QUESTION [8 upvotes]: A graph whose closure is the complete graph is Hamiltonian by the Bondy-Chvátal theorem, but I haven't found a polynomial algorithm for finding a Hamiltonian cycle in such a graph.  Is there one that we know of?
https://en.wikipedia.org/wiki/Bondy%E2%80%93Chv%C3%A1tal_theorem#Bondy.E2.80.93Chv.C3.A1tal_theorem
I don't see that class here http://www.graphclasses.org/classes/problem_Hamiltonian_cycle.html

REPLY [5 votes]: Tony's method works in the general case.  Let $e_1, e_2, \ldots, e_N$ be the edges that are added to $G$, in the order they are added, to make the complete graph.  Choose an arbitrary hamiltonian cycle in the complete graph.  Now remove $e_N$.  If $e_N$ was in the cycle, you can find a new cycle that avoids it by the cross-over technique that Tony explains. Then remove $e_{N-1}$ in the same way.  Continue until you removed all the edges that were added.  Since each step takes time $O(n)$, the total time is $O(nN)=O(n^3)$.  I don't believe this is best possible and wouldn't be surprised if this can be implemented in time $O(n^2)$.