Coxeter Decompositions of Hyperbolic Polygons

LetPbe a polygon on hyperbolic plane H2. A Coxeter decomposition of a polygonPis a non-trivial decomposition ofPinto finitely many Coxeter polygonsFi, such that any two polygonsF1andF2having a common side are symmetric with respect to this common side. In this paper we classify the Coxeter decompositions of triangles, quadrilaterals and ideal polygons.