Characterization of self-similar LabelSearch instances

Characterize the partial orders on finite labels for which LabelSearch becomes self-similar, thereby providing a satisfactory characterization that includes self-similar instances such as Maximal Neighborhood Search even though Maximal Neighborhood Search does not satisfy the third sufficient condition in Theorem self-similar-label.

Background

The paper introduces self-similar graph searches and proves sufficient conditions on the label partial order ensuring that LabelSearch is self-similar. The authors note that these conditions are not necessary: Maximal Neighborhood Search is self-similar despite violating the third condition. They therefore explicitly leave open the problem of finding a satisfactory characterization of all self-similar LabelSearch instances.