Simplicial Decompositions: Some New Aspects and Applications

Decompositions and unique representation theorems arising from them play a great role in algebra. There is, for instance, the decomposition induced by the Cartesian product. Another kind of decompositions having some formal similarities with algebraic factorizations is the simplicia1 decomposition (SD). The chapter explains the notion of simplicial decomposition. In addition, it reviews prime-graph decompositions. Triangulated graphs, along with some generalizations of the SD, are also discussed in the chapter; the two decomposition theorems described in the chapter are used to solve some coloring and connectivity problems. The separating simplices of a graph G have the property that none of them is separated by another one.