Unifying the representation of symmetric crossing families and weakly partitive families
Résumé
The family of non-trivial minimizers of a symmetric submodular function is symmetric crossing, namely it is closed under the complementation of any member and under the intersection of its crossing members. The family of modules of a graph is weakly partitive, namely it is closed under the intersection, union, and difference of its overlapping members. It is known that any symmetric crossing (resp. weakly partitive) family $F \subseteq 2^X$ has an $O(|X|)$ space representation. In a previous work we gave a general framework for representing any set family by a tree. This is a natural extension of the above mentioned result on symmetric crossing families. We here show how our framework also captures, in a non-trivial way, the above mentioned result on weakly partitive families. Among the consequences, this is the first result generalizing both the modular decomposition of a graph and the structural behaviour of the minimizers of a symmetric submodular function.