Arrangements of circles, the regions surrounded by them and labeled Poincaré-Reeb graphs

Abstract: We are interested in arrangements of circles and the regions surrounded by them. {\it Poincar\'e-Reeb graphs} have been fundamental and strong tools in studying shapes of regions surrounded by real algebraic curves, since around 2020. They are natural graphs the regions naturally collapse to and were first formulated by Sorea with several researchers. Studying shapes of such regions is one of fundamental studies in real algebraic geometry and combinatorics for example. This is surprisingly new and recently developing. Our study introduces labels on vertices and edges of such graphs encoding information of the circles where we concentrate on regions surrounded by circles. The author studied local changes of Poincar\'e-Reeb graphs by addition of circles under certain rules before and we discuss changes of new types. The author has started related studies motivated by singularity theory of real algebraic maps and found first that our regions are the images of natural real algebraic maps, generalizing natural projections of spheres.