Realizations of planar graphs as Poincar'e-Reeb graphs of refined algebraic domains

Abstract: Algebraic domains are regions in the plane surrounded by mutually disjoint non-singular real algebraic curves. Poincar'e-Reeb Graphs of them are graphs they naturally collapse: such graphs are formally formulated by Sorea, for example, around 2020. Their studies found that nicely embedded planar graphs are Poincar'e-Reeb graphs of some algebraic domains. These graphs are generic with respect to the projection to the horizontal axis. Problems, methods and results are elementary and natural and they apply natural approximations nicely for example. We present our new approach to extension of the result to a non-generic case and an answer. We first formulate generalized algebraic domains, surrounded by non-singular real algebraic curves which may intersect with normal crossings. Such domains and certain classes of them appear in related studies of graphs and regions surrounded by algebraic curves explicitly.