Branch and Bound algorithm on an integer problem When performing the branch and bound algorithm on an integer problem, multiple integer feasible solutions may be found.
Generally, is the second best integer feasible solution found during the branch and bound process
necessarily the second best integer solution of the Integer Problem??

No, a second-best solution might be pruned away and never encountered.  A simple modification to find the best two solutions is to keep a pair of solutions (instead of just one) as the incumbent, updating the incumbent whenever a solution is found that is better than the worse one.  This idea extends to the $k$ best solutions.
