Seidman (1983a) has suggested that the engineering concept of LS sets provides a good formaliza- tion of the intuitive network notion of a cohesive subset. Some desirable features that LS sets exhibit are that they are difficult to disconnect by removing edges, they are relatively dense within and isolated without, they have limited diameter, and individual members have more direct links to other members than to non-members. Unfortunately, this plethora of features means that LS sets occur only rarely in real data. It also means that they do not make good independent variables for structural analyses in which greater-than-expected in-group homogeneity is hypothe- sized with respect to some substantive dependent variable, because it is unclear which aspect of the LS set was responsible for the observed homogeneity. We discuss a variety of generalizations and relations of LS sets based on just a few of the properties possessed by LS sets. Some of these simpler models are drawn from the literature while others are introduced in this paper. One of the generalizations we introduce, called a lambda set, is based on the property that members of the set have greater edge connectivity with other members than with non-members. This property is shared by LS sets. Edge connectivity satisfies the axioms of an ultrametric similarity measure, and so LS sets and lambda sets are shown to correspond to a particular hierarchical clustering of the nodes in a network. Lambda sets are straightforward to compute, and we have made use of this fact to introduce a new algorithm for computing LS sets which runs an order of magnitude faster than the previous alternative.