Partially Broken Orientations of Eulerian Plane Graphs

It is well-known that every Eulerian plane graph G is face 2-colorable and admits an orientation which is an assignment of a direction to each edge of G such that incoming edges and outgoing edges appear alternately around any v∈V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in V(G)$$\end{document}; we say that such a vertex v has the alternate property, and that such an orientation is good. In this paper, we discuss orientations given to Eulerian plane graphs such that some specified vertices do not have the alternate property (while the others have the property), and give a characterization in terms of the radial graph of the Eulerian plane graph. Furthermore, for a given properly drawn graph on the plane, we discuss whether it has a good orientation or not.