  In this paper, we consider one-to-one matchings between two disjoint groups
of agents. Each agent has a preference over a subset of the agents in the other
group, and these preferences may contain ties. Strong stability is one of the
stability concepts in this setting. In this paper, we consider the following
variant of strong stability, and we consider computational complexity issues
for this solution concept. In our setting, we are given a subset of the agents
on one side. Then when an agent in this subset is not matched to any partner,
this agent cannot become a part of a blocking pair. In this paper, we first
prove that the problem of determining the existence of a stable matching in
this setting is NP-complete. Then we give two polynomial-time solvable cases of
our problem. Interestingly, one of our positive results gives a unified
approach to the strongly stable matching problem and the envy-free matching
problem.
