Complexity of recognizing equal unions in families of sets

A family of sets has the equal union property if there exist two nonempty d isjoint subfamilies having equal unions and has the full equal union proper ty if, in addition, all sets are included. Both recognition problems,are NP -complete even when restricted to families for which the cardinality of eve ry set is at most three. Both problems can be solved in polynomial time whe n restricted to families having a bounded number of sets with cardinality g reater than two. A corollary is that deciding if a graph has two disjoint e dge covers is in P. (C) 2000 Academic Press.