  Given an undirected graph, the stable set problem asks to determine the
cardinality of the largest subset of pairwise non-adjacent vertices. This value
is called the stability number of the graph, and its computation is an NP-hard
problem. In this paper, we focus on solving the stable set problem using the
D-wave quantum annealer. By formulating the problem as a quadratic
unconstrained binary optimization problem with the penalty method, we show that
the optimal value of this formulation is equal to the stability number of the
graph for certain penalty parameter values. However, D-Wave's quantum annealer
is not an exact solver, so the solutions may be far from the optimum and may
not even represent stable sets. To address this, we introduce a post-processing
procedure that identifies samples that could lead to improved solutions and
extracts stable sets from them. Additionally, we propose a so-called simple
CH-partitioning method to handle larger instances that cannot be embedded on
D-Wave's quantum processing unit. Finally, we investigate how different penalty
parameter values affect the solutions' quality. Extensive computational results
show that the post-processing procedure significantly improves the solution
quality, while the simple CH-partitioning method successfully extends our
approach to medium-size instances.
