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

TL;DR

This paper extends the understanding of how planar graphs can be realized as Poincaré-Reeb graphs of algebraic domains, including non-generic cases with intersecting boundary curves, advancing the theoretical framework in algebraic and geometric graph representations.

Contribution

It introduces a generalized concept of algebraic domains with intersecting boundary curves and extends existing results to non-generic cases, providing new insights into graph realizations.

Findings

Extended the class of algebraic domains to include intersecting boundary curves.

Provided a theoretical framework for non-generic Poincaré-Reeb graph realizations.

Connected algebraic domain theory with graph embedding problems.

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.