# Representation theorems for two set families and applications to combinatorial decompositions

Abstract : A set family $F \subseteq 2^X$ is partitive crossing if it is close under the union, the intersection, and the difference of its crossing members; it is a union-difference family if closed under the union and the difference of its overlapping members. In both cases, the cardinality of $F$ is potentially in $O(2^{|X|})$, and the total cardinality of its members even higher. We give a linear $O(|X|)$ and a quadratic $O(|X|^2)$ space representation based on a canonical tree for any partitive crossing family and union-difference family, respectively. As an application of this framework we obtain a unique digraph decomposition and a unique decomposition of $2-$structure. Both of them not only captures, but also is strictly more powerful than the well-studied modular decomposition and clan decomposition. Polynomial time decomposition algorithms for both cases are described.
Document type :
Conference papers

https://hal.archives-ouvertes.fr/hal-00555315
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Submitted on : Thursday, January 13, 2011 - 1:35:51 AM
Last modification on : Friday, August 5, 2022 - 10:32:03 AM

### Identifiers

• HAL Id : hal-00555315, version 1

### Citation

Binh-Minh Bui-Xuan, Michel Habib, Michaël Rao. Representation theorems for two set families and applications to combinatorial decompositions. International Conference on Relations, Orders and Graphs: Interaction with Computer Science (ROGICS'08), 2008, Tunisia. pp.532-546. ⟨hal-00555315⟩

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