Survey on Decomposition of Multiple Coverings [chapter]

János Pach, Dömötör Pálvölgyi, Géza Tóth
2013 Bolyai Society Mathematical Studies  
>IJH=?J The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decomposability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline
more » ... survey, we outline the most important results, methods, and questions. 1 Cover-decomposability and the sensor cover problem Let P = { P i | i ∈ I } be a collection of sets in R d . We say that P is an m-fold covering if every point of R d is contained in at least m members of P. The largest such m is called the thickness of the covering. A 1-fold covering is simply called a covering. To formulate the central question of this survey succinctly, we need a denition. Denition 1.1. A planar set P is said to be cover-decomposable if there exists a (minimal) constant m = m(P ) such that every m-fold covering of the plane with translates of P can be decomposed into two coverings. Note that the above term is slightly misleading: we decompose (partition) not the set P , but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P.
doi:10.1007/978-3-642-41498-5_9 fatcat:br5j6tjlbbfkxot5oo2ywiiiya