In the last decade, graph-cut optimization has been popular for a variety of pixel labeling problems. Typically graph-cut methods are used to incorporate a smoothness prior on a labeling. Recently several methods incorporated ordering constraints on labels for the application of object segmentation. An example of an ordering constraint is prohibiting a pixel with a "car wheel" label to be above a pixel with a "car roof" label. We observe that the commonly used graph-cut based -expansion is more likely to get stuck in a local minimum when ordering constraints are used. For certain models with ordering constraints, we develop new graph-cut moves which we call order-preserving moves. Order-preserving moves act on all labels, unlike expansion. Although the global minimum is still not guaranteed, optimization with order-preserving moves performs significantly better than -expansion. We evaluate orderpreserving moves for the geometric class scene labeling (introduced by ...