Recognizing circle graphs in polynomial time
Journal of the ACM (JACM)
Centroids, representations, and submodular flows
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
Algorithmic aspects of a general modular decomposition theory
Discrete Applied Mathematics
A representation theorem for union-difference families and application
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Tree-representation of set families and applications to combinatorial decompositions
European Journal of Combinatorics
Tree-representation of set families and applications to combinatorial decompositions
European Journal of Combinatorics
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A set family is a collection of sets over a universe. If a set family satisfies certain closure properties then it admits an efficient representation of its members by labeled trees. The size of the tree is proportional to the size of the universe, whereas the number of set family members can be exponential. Computing such efficient representations is an important task in algorithm design. Set families are usually not given explicitly (by listing their members) but represented implicitly. We consider the problem of efficiently computing tree representations of set families. Assuming the existence of efficient algorithms for solving the Membership and Separation problems, we prove that if a set family satisfies weak closure properties then there exists an efficient algorithm for computing a tree representation of the set family. The running time of the algorithm will mainly depend on the running times of the algorithms for the two basic problems. Our algorithm generalizes several previous results and provides a unified approach to the computation for a large class of decompositions of graphs. We also introduce a decomposition notion for directed graphs which has no undirected analogue. We show that the results of the first part of the paper are applicable to this new decomposition. Finally, we give efficient algorithms for the two basic problems and obtain an O(n3)-time algorithm for computing a tree representation.